diff --git a/lib/live_book/application.ex b/lib/live_book/application.ex
index a743adf20..89bb142a8 100644
--- a/lib/live_book/application.ex
+++ b/lib/live_book/application.ex
@@ -13,6 +13,8 @@ defmodule LiveBook.Application do
       {Phoenix.PubSub, name: LiveBook.PubSub},
       # Start the supervisor dynamically managing sessions
       LiveBook.SessionSupervisor,
+      # Start the supervisor dynamically spawning evaluator servers
+      LiveBook.EvaluatorSupervisor,
       # Start the Endpoint (http/https)
       LiveBookWeb.Endpoint
     ]
diff --git a/lib/live_book/evaluator.ex b/lib/live_book/evaluator.ex
index 08cf4cac5..1375c4a00 100644
--- a/lib/live_book/evaluator.ex
+++ b/lib/live_book/evaluator.ex
@@ -11,7 +11,7 @@ defmodule LiveBook.Evaluator do
   # where the evaluation happens, as otherwise we would have to
   # send them between processes, effectively copying potentially large data.
 
-  use GenServer
+  use GenServer, restart: :temporary
 
   alias LiveBook.Evaluator
 
@@ -46,23 +46,19 @@ defmodule LiveBook.Evaluator do
   end
 
   @doc """
-  Synchronously parses and evaluates the given code.
+  Asynchronously parses and evaluates the given code.
 
   Any exceptions are captured, in which case this method returns an error.
 
   The evaluator stores the resulting binding and environment under `ref`.
   Any subsequent calls may specify `prev_ref` pointing to a previous evaluation,
   in which case the corresponding binding and environment are used during evaluation.
+
+  Evaluation response is sent to the process identified by `send_to` as `{:evaluation_response, ref, response}`.
   """
-  @spec evaluate_code(t(), String.t(), ref(), ref()) :: evaluation_response()
-  def evaluate_code(evaluator, code, ref, prev_ref \\ :initial) when ref != :initial do
-    response = GenServer.call(evaluator, {:evaluate_code, code, ref, prev_ref}, :infinity)
-
-    if response == :invalid_prev_ref do
-      raise ArgumentError, message: "invalid reference to previous evaluation: #{prev_ref}"
-    end
-
-    response
+  @spec evaluate_code(t(), pid(), String.t(), ref(), ref()) :: :ok
+  def evaluate_code(evaluator, send_to, code, ref, prev_ref \\ :initial) when ref != :initial do
+    GenServer.cast(evaluator, {:evaluate_code, send_to, code, ref, prev_ref})
   end
 
   @doc """
@@ -99,29 +95,26 @@ defmodule LiveBook.Evaluator do
   end
 
   @impl true
-  def handle_call({:evaluate_code, code, ref, prev_ref}, {from, _}, state) do
-    case Map.fetch(state.contexts, prev_ref) do
-      :error ->
-        {:reply, :invalid_prev_ref, state}
+  def handle_cast({:evaluate_code, send_to, code, ref, prev_ref}, state) do
+    Evaluator.IOProxy.configure(state.io_proxy, send_to, ref)
 
-      {:ok, context} ->
-        Evaluator.IOProxy.configure(state.io_proxy, from, ref)
+    context = Map.get(state.contexts, prev_ref, state.contexts.initial)
 
-        case eval(code, context.binding, context.env) do
-          {:ok, result, binding, env} ->
-            result_context = %{binding: binding, env: env}
-            new_contexts = Map.put(state.contexts, ref, result_context)
-            new_state = %{state | contexts: new_contexts}
+    case eval(code, context.binding, context.env) do
+      {:ok, result, binding, env} ->
+        result_context = %{binding: binding, env: env}
+        new_contexts = Map.put(state.contexts, ref, result_context)
+        new_state = %{state | contexts: new_contexts}
 
-            {:reply, {:ok, result}, new_state}
+        send(send_to, {:evaluator_response, ref, {:ok, result}})
+        {:noreply, new_state}
 
-          {:error, kind, error, stacktrace} ->
-            {:reply, {:error, kind, error, stacktrace}, state}
-        end
+      {:error, kind, error, stacktrace} ->
+        send(send_to, {:evaluator_response, ref, {:error, kind, error, stacktrace}})
+        {:noreply, state}
     end
   end
 
-  @impl true
   def handle_cast({:forget_evaluation, ref}, state) do
     new_state = %{state | contexts: Map.delete(state.contexts, ref)}
     {:noreply, new_state}
diff --git a/lib/live_book/evaluator_supervisor.ex b/lib/live_book/evaluator_supervisor.ex
new file mode 100644
index 000000000..ac9ddea7f
--- /dev/null
+++ b/lib/live_book/evaluator_supervisor.ex
@@ -0,0 +1,43 @@
+defmodule LiveBook.EvaluatorSupervisor do
+  @moduledoc false
+
+  # Supervisor responsible for dynamically spawning
+  # and terminating terminator server processes.
+
+  use DynamicSupervisor
+
+  alias LiveBook.Evaluator
+
+  @name __MODULE__
+
+  def start_link(opts \\ []) do
+    DynamicSupervisor.start_link(__MODULE__, opts, name: @name)
+  end
+
+  @impl true
+  def init(_opts) do
+    DynamicSupervisor.init(strategy: :one_for_one)
+  end
+
+  @doc """
+  Spawns a new evaluator.
+  """
+  @spec start_evaluator() :: {:ok, Evaluator.t()} | {:error, any()}
+  def start_evaluator() do
+    case DynamicSupervisor.start_child(@name, Evaluator) do
+      {:ok, pid} -> {:ok, pid}
+      {:ok, pid, _} -> {:ok, pid}
+      :ignore -> {:error, :ignore}
+      {:error, reason} -> {:error, reason}
+    end
+  end
+
+  @doc """
+  Terminates the given evaluator.
+  """
+  @spec terminate_evaluator(Evaluator.t()) :: :ok
+  def terminate_evaluator(evaluator) do
+    DynamicSupervisor.terminate_child(@name, evaluator)
+    :ok
+  end
+end
diff --git a/lib/live_book/notebook.ex b/lib/live_book/notebook.ex
index d7f927339..011f5fbf1 100644
--- a/lib/live_book/notebook.ex
+++ b/lib/live_book/notebook.ex
@@ -2,7 +2,7 @@ defmodule LiveBook.Notebook do
   @moduledoc """
   Data structure representing a notebook.
 
-  A notebook it's just the representation and roughly
+  A notebook is just the representation and roughly
   maps to a file that the user can edit.
   A notebook *session* is a living process that holds a specific
   notebook instance and allows users to collaboratively apply
@@ -13,7 +13,7 @@ defmodule LiveBook.Notebook do
 
   defstruct [:name, :version, :sections, :metadata]
 
-  alias LiveBook.Notebook.Section
+  alias LiveBook.Notebook.{Section, Cell}
 
   @type t :: %__MODULE__{
           name: String.t(),
@@ -36,4 +36,135 @@ defmodule LiveBook.Notebook do
       metadata: %{}
     }
   end
+
+  @doc """
+  Finds notebook section by id.
+  """
+  @spec fetch_section(t(), Section.id()) :: {:ok, Section.t()} | :error
+  def fetch_section(notebook, section_id) do
+    Enum.find_value(notebook.sections, :error, fn section ->
+      section.id == section_id && {:ok, section}
+    end)
+  end
+
+  @doc """
+  Finds notebook cell by `id` and the corresponding section.
+  """
+  @spec fetch_cell_and_section(t(), Cell.section_id()) :: {:ok, Cell.t(), Section.t()} | :error
+  def fetch_cell_and_section(notebook, cell_id) do
+    for(
+      section <- notebook.sections,
+      cell <- section.cells,
+      cell.id == cell_id,
+      do: {cell, section}
+    )
+    |> case do
+      [{cell, section}] -> {:ok, cell, section}
+      [] -> :error
+    end
+  end
+
+  @doc """
+  Inserts `section` at the given `index`.
+  """
+  @spec insert_section(t(), integer(), Section.t()) :: t()
+  def insert_section(notebook, index, section) do
+    sections = List.insert_at(notebook.sections, index, section)
+
+    %{notebook | sections: sections}
+  end
+
+  @doc """
+  Inserts `cell` at the given `index` within section identified by `section_id`.
+  """
+  @spec insert_cell(t(), Section.id(), integer(), Cell.t()) :: t()
+  def insert_cell(notebook, section_id, index, cell) do
+    sections =
+      Enum.map(notebook.sections, fn section ->
+        if section.id == section_id do
+          %{section | cells: List.insert_at(section.cells, index, cell)}
+        else
+          section
+        end
+      end)
+
+    %{notebook | sections: sections}
+  end
+
+  @doc """
+  Deletes section with the given id.
+  """
+  @spec delete_section(t(), Section.id()) :: t()
+  def delete_section(notebook, section_id) do
+    sections = Enum.reject(notebook.sections, &(&1.id == section_id))
+
+    %{notebook | sections: sections}
+  end
+
+  @doc """
+  Deletes cell with the given id.
+  """
+  @spec delete_cell(t(), Cell.id()) :: t()
+  def delete_cell(notebook, cell_id) do
+    sections =
+      Enum.map(notebook.sections, fn section ->
+        %{section | cells: Enum.reject(section.cells, &(&1.id == cell_id))}
+      end)
+
+    %{notebook | sections: sections}
+  end
+
+  @doc """
+  Updates cell with the given function.
+  """
+  @spec update_cell(t(), Cell.id(), (Cell.t() -> Cell.t())) :: t()
+  def update_cell(notebook, cell_id, fun) do
+    sections =
+      Enum.map(notebook.sections, fn section ->
+        cells =
+          Enum.map(section.cells, fn cell ->
+            if cell.id == cell_id, do: fun.(cell), else: cell
+          end)
+
+        %{section | cells: cells}
+      end)
+
+    %{notebook | sections: sections}
+  end
+
+  @doc """
+  Returns a list of Elixir cells that the given cell depends on.
+
+  The cells are ordered starting from the most direct parent.
+  """
+  @spec parent_cells(t(), Cell.id()) :: list(Cell.t())
+  def parent_cells(notebook, cell_id) do
+    with {:ok, _, section} <- LiveBook.Notebook.fetch_cell_and_section(notebook, cell_id) do
+      # A cell depends on all previous cells within the same section.
+      section.cells
+      |> Enum.filter(&(&1.type == :elixir))
+      |> Enum.take_while(&(&1.id != cell_id))
+      |> Enum.reverse()
+    else
+      _ -> []
+    end
+  end
+
+  @doc """
+  Returns a list of Elixir cells that depend on the given cell.
+
+  The cells are ordered starting from the most direct child.
+  """
+  @spec child_cells(t(), Cell.id()) :: list(Cell.t())
+  def child_cells(notebook, cell_id) do
+    with {:ok, _, section} <- LiveBook.Notebook.fetch_cell_and_section(notebook, cell_id) do
+      # A cell affects all the cells below it within the same section.
+      section.cells
+      |> Enum.filter(&(&1.type == :elixir))
+      |> Enum.reverse()
+      |> Enum.take_while(&(&1.id != cell_id))
+    else
+      _ -> []
+    end
+  end
 end
diff --git a/lib/live_book/notebook/cell.ex b/lib/live_book/notebook/cell.ex
index 0c7b674c1..8f3f84f32 100644
--- a/lib/live_book/notebook/cell.ex
+++ b/lib/live_book/notebook/cell.ex
@@ -11,12 +11,12 @@ defmodule LiveBook.Notebook.Cell do
 
   alias LiveBook.Utils
 
-  @type cell_id :: Utils.id()
-  @type cell_type :: :markdown | :elixir
+  @type id :: Utils.id()
+  @type type :: :markdown | :elixir
 
   @type t :: %__MODULE__{
-          id: cell_id(),
-          type: cell_type(),
+          id: id(),
+          type: type(),
           source: String.t(),
           # TODO: expand on this
           outputs: list(),
@@ -26,7 +26,7 @@ defmodule LiveBook.Notebook.Cell do
   @doc """
   Returns an empty cell of the given type.
   """
-  @spec new(cell_type()) :: t()
+  @spec new(type()) :: t()
   def new(type) do
     %__MODULE__{
       id: Utils.random_id(),
diff --git a/lib/live_book/notebook/section.ex b/lib/live_book/notebook/section.ex
index 7fe788b15..ae7bf0b2e 100644
--- a/lib/live_book/notebook/section.ex
+++ b/lib/live_book/notebook/section.ex
@@ -11,10 +11,10 @@ defmodule LiveBook.Notebook.Section do
   alias LiveBook.Notebook.Cell
   alias LiveBook.Utils
 
-  @type section_id :: Utils.id()
+  @type id :: Utils.id()
 
   @type t :: %__MODULE__{
-          id: section_id(),
+          id: id(),
           name: String.t(),
           cells: list(Cell.t()),
           metadata: %{atom() => term()}
diff --git a/lib/live_book/session.ex b/lib/live_book/session.ex
index a8c648a2c..14c0a1972 100644
--- a/lib/live_book/session.ex
+++ b/lib/live_book/session.ex
@@ -7,13 +7,28 @@ defmodule LiveBook.Session do
   # as a source of truth that multiple clients talk to.
   # Receives update requests from the clients and notifies
   # them of any changes applied to the notebook.
+  #
+  # The core concept is the `Data` structure
+  # to which we can apply reproducible opreations.
+  # See `Data` for more information.
 
   use GenServer, restart: :temporary
 
+  alias LiveBook.Session.Data
+  alias LiveBook.{Evaluator, EvaluatorSupervisor, Utils, Notebook}
+  alias LiveBook.Notebook.{Cell, Section}
+
+  @type state :: %{
+          session_id: id(),
+          data: Data.t(),
+          evaluators: %{Section.t() => Evaluator.t()},
+          client_pids: list(pid())
+        }
+
   @typedoc """
   An id assigned to every running session process.
   """
-  @type session_id :: LiveBook.Utils.id()
+  @type id :: Utils.id()
 
   ## API
 
@@ -21,7 +36,7 @@ defmodule LiveBook.Session do
   Starts the server process and registers it globally using the `:global` module,
   so that it's identifiable by the given id.
   """
-  @spec start_link(session_id()) :: GenServer.on_start()
+  @spec start_link(id()) :: GenServer.on_start()
   def start_link(session_id) do
     GenServer.start_link(__MODULE__, [session_id: session_id], name: name(session_id))
   end
@@ -30,10 +45,61 @@ defmodule LiveBook.Session do
     {:global, {:session, session_id}}
   end
 
+  @doc """
+  Registers a session client, so that it receives updates from the server.
+
+  The client process is automatically unregistered when it terminates.
+  """
+  @spec register_client(id(), pid()) :: :ok
+  def register_client(session_id, pid) do
+    GenServer.cast(name(session_id), {:register_client, pid})
+  end
+
+  @doc """
+  Asynchronously sends section insertion request to the server.
+  """
+  @spec insert_section(id(), non_neg_integer()) :: :ok
+  def insert_section(session_id, index) do
+    GenServer.cast(name(session_id), {:insert_section, index})
+  end
+
+  @doc """
+  Asynchronously sends cell insertion request to the server.
+  """
+  @spec insert_cell(id(), Section.id(), non_neg_integer(), Cell.type()) ::
+          :ok
+  def insert_cell(session_id, section_id, index, type) do
+    GenServer.cast(name(session_id), {:insert_cell, section_id, index, type})
+  end
+
+  @doc """
+  Asynchronously sends section deletion request to the server.
+  """
+  @spec delete_section(id(), Section.id()) :: :ok
+  def delete_section(session_id, section_id) do
+    GenServer.cast(name(session_id), {:delete_section, section_id})
+  end
+
+  @doc """
+  Asynchronously sends cell deletion request to the server.
+  """
+  @spec delete_cell(id(), Cell.id()) :: :ok
+  def delete_cell(session_id, cell_id) do
+    GenServer.cast(name(session_id), {:delete_cell, cell_id})
+  end
+
+  @doc """
+  Asynchronously sends cell evaluation request to the server.
+  """
+  @spec queue_cell_evaluation(id(), Cell.id()) :: :ok
+  def queue_cell_evaluation(session_id, cell_id) do
+    GenServer.cast(name(session_id), {:queue_cell_evaluation, cell_id})
+  end
+
   @doc """
   Synchronously stops the server.
   """
-  @spec stop(session_id()) :: :ok
+  @spec stop(id()) :: :ok
   def stop(session_id) do
     GenServer.stop(name(session_id))
   end
@@ -41,7 +107,163 @@ defmodule LiveBook.Session do
   ## Callbacks
 
   @impl true
-  def init(session_id: _id) do
-    {:ok, %{}}
+  def init(session_id: session_id) do
+    {:ok,
+     %{
+       session_id: session_id,
+       data: Data.new(),
+       evaluators: %{},
+       client_pids: []
+     }}
+  end
+
+  @impl true
+  def handle_cast({:register_client, pid}, state) do
+    Process.monitor(pid)
+    {:noreply, %{state | client_pids: [pid | state.client_pids]}}
+  end
+
+  def handle_cast({:insert_section, index}, state) do
+    # Include new id in the operation, so it's reproducible
+    operation = {:insert_section, index, Utils.random_id()}
+    handle_operation(state, operation)
+  end
+
+  def handle_cast({:insert_cell, section_id, index, type}, state) do
+    # Include new id in the operation, so it's reproducible
+    operation = {:insert_cell, section_id, index, type, Utils.random_id()}
+    handle_operation(state, operation)
+  end
+
+  def handle_cast({:delete_section, section_id}, state) do
+    operation = {:delete_section, section_id}
+
+    handle_operation(state, operation, fn new_state ->
+      delete_section_evaluator(new_state, section_id)
+    end)
+  end
+
+  def handle_cast({:delete_cell, cell_id}, state) do
+    operation = {:delete_cell, cell_id}
+    handle_operation(state, operation)
+  end
+
+  def handle_cast({:queue_cell_evaluation, cell_id}, state) do
+    operation = {:queue_cell_evaluation, cell_id}
+
+    handle_operation(state, operation, fn new_state ->
+      maybe_trigger_evaluations(state, new_state)
+    end)
+  end
+
+  @impl true
+  def handle_info({:DOWN, _, :process, pid, _}, state) do
+    {:noreply, %{state | client_pids: List.delete(state.client_pids, pid)}}
+  end
+
+  def handle_info({:evaluator_stdout, cell_id, string}, state) do
+    operation = {:add_cell_evaluation_stdout, cell_id, string}
+    handle_operation(state, operation)
+  end
+
+  def handle_info({:evaluator_response, cell_id, response}, state) do
+    operation = {:add_cell_evaluation_response, cell_id, response}
+
+    handle_operation(state, operation, fn new_state ->
+      maybe_trigger_evaluations(state, new_state)
+    end)
+  end
+
+  # ---
+
+  # Given any opeation on `Data`, the process does the following:
+  #
+  #   * broadcasts the operation to all clients immediately,
+  #     so that they can update their local `Data`
+  #   * applies the operation to own local `Data`
+  #   * optionally performs a relevant task (e.g. starts cell evaluation),
+  #     to reflect the new `Data`
+  #
+  defp handle_operation(state, operation) do
+    handle_operation(state, operation, fn state -> state end)
+  end
+
+  defp handle_operation(state, operation, handle_new_state) do
+    broadcast_operation(state.session_id, operation)
+
+    case Data.apply_operation(state.data, operation) do
+      {:ok, new_data} ->
+        new_state = %{state | data: new_data}
+        {:noreply, handle_new_state.(new_state)}
+
+      :error ->
+        {:noreply, state}
+    end
+  end
+
+  defp broadcast_operation(session_id, operation) do
+    message = {:operation, operation}
+    Phoenix.PubSub.broadcast(LiveBook.PubSub, "sessions:#{session_id}", message)
+  end
+
+  # Compares sections in the old and new state and if a new cell
+  # has been marked as evaluating it triggers the actual evaluation task.
+  defp maybe_trigger_evaluations(old_state, new_state) do
+    Enum.reduce(new_state.data.notebook.sections, new_state, fn section, state ->
+      case {Data.get_evaluating_cell_id(old_state.data, section.id),
+            Data.get_evaluating_cell_id(new_state.data, section.id)} do
+        {_, nil} ->
+          # No cell to evaluate
+          state
+
+        {cell_id, cell_id} ->
+          # The evaluating cell hasn't changed, so it must be already evaluating
+          state
+
+        {_, cell_id} ->
+          # The evaluating cell changed, so we trigger the evaluation to reflect that
+          trigger_evaluation(state, cell_id)
+      end
+    end)
+  end
+
+  defp trigger_evaluation(state, cell_id) do
+    notebook = state.data.notebook
+    {:ok, cell, section} = Notebook.fetch_cell_and_section(notebook, cell_id)
+    {state, evaluator} = get_section_evaluator(state, section.id)
+    %{source: source} = cell
+
+    prev_ref =
+      case Notebook.parent_cells(notebook, cell_id) do
+        [parent | _] -> parent.id
+        [] -> :initial
+      end
+
+    Evaluator.evaluate_code(evaluator, self(), source, cell_id, prev_ref)
+
+    state
+  end
+
+  defp get_section_evaluator(state, section_id) do
+    case Map.fetch(state.evaluators, section_id) do
+      {:ok, evaluator} ->
+        {state, evaluator}
+
+      :error ->
+        {:ok, evaluator} = EvaluatorSupervisor.start_evaluator()
+        state = %{state | evaluators: Map.put(state.evaluators, section_id, evaluator)}
+        {state, evaluator}
+    end
+  end
+
+  defp delete_section_evaluator(state, section_id) do
+    case Map.fetch(state.evaluators, section_id) do
+      {:ok, evaluator} ->
+        EvaluatorSupervisor.terminate_evaluator(evaluator)
+        %{state | evaluators: Map.delete(state.evaluators, section_id)}
+
+      :error ->
+        state
+    end
   end
 end
diff --git a/lib/live_book/session/data.ex b/lib/live_book/session/data.ex
new file mode 100644
index 000000000..d033b737b
--- /dev/null
+++ b/lib/live_book/session/data.ex
@@ -0,0 +1,361 @@
+defmodule LiveBook.Session.Data do
+  @moduledoc false
+
+  # A structure with shared session data.
+  #
+  # In some sense this structure is a `Notebook` decorated
+  # with all the emphemeral session data.
+  #
+  # The data is kept both in the `Session` process and all the client processes.
+  # All changes go to the `Session` process first to introduce linearity
+  # and then are broadcasted to the clients, hence every client
+  # receives changes in the same order.
+  # Upon receiving a change message, every process applies
+  # the change to the locally stored `Data`. In this way the local `Data`
+  # stays the same in all processes, while the messages are minimal.
+
+  defstruct [
+    :notebook,
+    :path,
+    :section_infos,
+    :cell_infos,
+    :deleted_sections,
+    :deleted_cells
+  ]
+
+  alias LiveBook.{Notebook, Evaluator}
+  alias LiveBook.Notebook.{Cell, Section}
+
+  @type t :: %__MODULE__{
+          notebook: Notebook.t(),
+          path: nil | String.t(),
+          section_infos: %{Section.id() => section_info()},
+          cell_infos: %{Cell.id() => cell_info()},
+          deleted_sections: list(Section.t()),
+          deleted_cells: list(Cell.t())
+        }
+
+  @type section_info :: %{
+          evaluating_cell_id: Cell.id(),
+          evaluation_queue: list(Cell.id())
+        }
+
+  @type cell_info :: %{
+          status: cell_status(),
+          revision: non_neg_integer(),
+          # TODO: specify it's a list of deltas, once defined
+          deltas: list(),
+          evaluated_at: DateTime.t()
+        }
+
+  @type cell_status :: :fresh | :queued | :evaluating | :evaluated | :stale
+
+  @type index :: non_neg_integer()
+
+  @type operation ::
+          {:insert_section, index(), Section.id()}
+          | {:insert_cell, Section.id(), index(), Cell.type(), Cell.id()}
+          | {:delete_section, Section.id()}
+          | {:delete_cell, Cell.id()}
+          | {:queue_cell_evaluation, Cell.id()}
+          | {:add_cell_evaluation_stdout, Cell.id(), String.t()}
+          | {:add_cell_evaluation_response, Cell.id(), Evaluator.evaluation_response()}
+
+  @doc """
+  Returns a fresh notebook session state.
+  """
+  @spec new() :: t()
+  def new() do
+    %__MODULE__{
+      notebook: Notebook.new(),
+      path: nil,
+      section_infos: %{},
+      cell_infos: %{},
+      deleted_sections: [],
+      deleted_cells: []
+    }
+  end
+
+  @doc """
+  Applies the change specified by `operation` to the given session `data`.
+
+  All operations are reproducible (i.e. this function is pure),
+  so provided all processes have the same session data
+  they can individually apply any given operation and end up in the same state.
+
+  An operation only applies changes to the structure, but it doesn't trigger
+  any actual processing. It's the responsibility of the session process to ensure
+  the system reflects the new structure. For instance, when a new cell is marked
+  as evaluating, the session process should take care of triggering actual evaluation.
+
+  Returns `{:ok, data}` on correct application or `:error` if the operation
+  is not valid. The `:error` is generally expected given the collaborative
+  nature of sessions. For example if there are simultaneous deletion
+  and evaluation operations on the same cell, we may perform delete first,
+  in which case the evaluation is no longer valid (there's no cell with the given id).
+  By returning `:error` we simply notify the caller that no changes were applied,
+  so any related actions can be ignored.
+  """
+  @spec apply_operation(t(), operation()) :: {:ok, t()} | :error
+  def apply_operation(data, operation)
+
+  def apply_operation(data, {:insert_section, index, id}) do
+    section = %{Section.new() | id: id}
+
+    data
+    |> insert_section(index, section)
+    |> wrap_ok()
+  end
+
+  def apply_operation(data, {:insert_cell, section_id, index, type, id}) do
+    with {:ok, _section} <- Notebook.fetch_section(data.notebook, section_id) do
+      cell = %{Cell.new(type) | id: id}
+
+      data
+      |> insert_cell(section_id, index, cell)
+      |> wrap_ok()
+    end
+  end
+
+  def apply_operation(data, {:delete_section, id}) do
+    with {:ok, section} <- Notebook.fetch_section(data.notebook, id),
+         # If a cell within this section is being evaluated, it should be cancelled first
+         nil <- data.section_infos[section.id].evaluating_cell_id do
+      data
+      |> delete_section(section)
+      |> wrap_ok()
+    else
+      _ -> :error
+    end
+  end
+
+  def apply_operation(data, {:delete_cell, id}) do
+    with {:ok, cell, section} <- Notebook.fetch_cell_and_section(data.notebook, id),
+         # If the cell is being evaluated, it should be cancelled first
+         false <- data.cell_infos[cell.id].status == :evaluating do
+      data
+      |> unqueue_cell_evaluation_if_any(cell, section)
+      |> mark_dependent_cells_as_stale(cell)
+      |> delete_cell(cell)
+      |> wrap_ok()
+    else
+      _ -> :error
+    end
+  end
+
+  def apply_operation(data, {:queue_cell_evaluation, id}) do
+    with {:ok, cell, section} <- Notebook.fetch_cell_and_section(data.notebook, id),
+         :elixir <- cell.type,
+         false <- data.cell_infos[cell.id].status in [:queued, :evaluating] do
+      prerequisites_queue = fresh_parent_cells_queue(data, cell)
+
+      data
+      |> reduce(prerequisites_queue, &queue_cell_evaluation(&1, &2, section))
+      |> queue_cell_evaluation(cell, section)
+      |> maybe_evaluate_queued()
+      |> wrap_ok()
+    else
+      _ -> :error
+    end
+  end
+
+  def apply_operation(data, {:add_cell_evaluation_stdout, id, string}) do
+    with {:ok, cell, _} <- Notebook.fetch_cell_and_section(data.notebook, id) do
+      data
+      |> add_cell_evaluation_stdout(cell, string)
+      |> wrap_ok()
+    end
+  end
+
+  def apply_operation(data, {:add_cell_evaluation_response, id, response}) do
+    with {:ok, cell, section} <- Notebook.fetch_cell_and_section(data.notebook, id) do
+      data
+      |> add_cell_evaluation_response(cell, response)
+      |> finish_cell_evaluation(cell, section)
+      |> mark_dependent_cells_as_stale(cell)
+      |> maybe_evaluate_queued()
+      |> wrap_ok()
+    end
+  end
+
+  # ===
+
+  defp insert_section(data, index, section) do
+    data
+    |> set!(
+      notebook: Notebook.insert_section(data.notebook, index, section),
+      section_infos: Map.put(data.section_infos, section.id, new_section_info())
+    )
+  end
+
+  defp insert_cell(data, section_id, index, cell) do
+    data
+    |> set!(
+      notebook: Notebook.insert_cell(data.notebook, section_id, index, cell),
+      cell_infos: Map.put(data.cell_infos, cell.id, new_cell_info())
+    )
+  end
+
+  defp delete_section(data, section) do
+    data
+    |> set!(
+      notebook: Notebook.delete_section(data.notebook, section.id),
+      section_infos: Map.delete(data.section_infos, section.id),
+      deleted_sections: [section | data.deleted_sections]
+    )
+    |> reduce(section.cells, &delete_cell_info/2)
+  end
+
+  defp delete_cell(data, cell) do
+    data
+    |> set!(
+      notebook: Notebook.delete_cell(data.notebook, cell.id),
+      deleted_cells: [cell | data.deleted_cells]
+    )
+    |> delete_cell_info(cell)
+  end
+
+  defp delete_cell_info(data, cell) do
+    data
+    |> set!(cell_infos: Map.delete(data.cell_infos, cell.id))
+  end
+
+  defp queue_cell_evaluation(data, cell, section) do
+    data
+    |> update_section_info!(section.id, fn section ->
+      %{section | evaluation_queue: section.evaluation_queue ++ [cell.id]}
+    end)
+    |> set_cell_info!(cell.id, status: :queued)
+  end
+
+  defp unqueue_cell_evaluation_if_any(data, cell, section) do
+    data
+    |> update_section_info!(section.id, fn section ->
+      %{section | evaluation_queue: List.delete(section.evaluation_queue, cell.id)}
+    end)
+    |> set_cell_info!(cell.id, status: :stale)
+  end
+
+  defp add_cell_evaluation_stdout(data, _cell, _string) do
+    data
+    |> set!(
+      # TODO: add stdout to cell outputs
+      notebook: data.notebook
+    )
+  end
+
+  defp add_cell_evaluation_response(data, _cell, _response) do
+    data
+    |> set!(
+      # TODO: add result to outputs
+      notebook: data.notebook
+    )
+  end
+
+  defp finish_cell_evaluation(data, cell, section) do
+    data
+    |> set_cell_info!(cell.id, status: :evaluated, evaluated_at: DateTime.utc_now())
+    |> set_section_info!(section.id, evaluating_cell_id: nil)
+  end
+
+  defp mark_dependent_cells_as_stale(data, cell) do
+    invalidated_cells = evaluated_child_cells(data, cell)
+
+    data
+    |> reduce(invalidated_cells, &set_cell_info!(&1, &2.id, status: :stale))
+  end
+
+  defp fresh_parent_cells_queue(data, cell) do
+    data.notebook
+    |> Notebook.parent_cells(cell.id)
+    |> Enum.filter(fn parent -> data.cell_infos[parent.id].status == :fresh end)
+    |> Enum.reverse()
+  end
+
+  defp evaluated_child_cells(data, cell) do
+    data.notebook
+    |> Notebook.child_cells(cell.id)
+    # Mark only evaluted cells as stale
+    |> Enum.filter(fn cell -> data.cell_infos[cell.id].status == :evaluated end)
+  end
+
+  # If there are idle sections with non-empty evaluation queue,
+  # the next queued cell for evaluation.
+  defp maybe_evaluate_queued(data) do
+    Enum.reduce(data.notebook.sections, data, fn section, data ->
+      case data.section_infos[section.id] do
+        %{evaluating_cell_id: nil, evaluation_queue: [id | ids]} ->
+          data
+          |> set!(notebook: Notebook.update_cell(data.notebook, id, &%{&1 | outputs: []}))
+          |> set_cell_info!(id, status: :evaluating)
+          |> set_section_info!(section.id, evaluating_cell_id: id, evaluation_queue: ids)
+
+        _ ->
+          data
+      end
+    end)
+  end
+
+  defp wrap_ok(value), do: {:ok, value}
+
+  defp new_section_info() do
+    %{
+      evaluating_cell_id: nil,
+      evaluation_queue: []
+    }
+  end
+
+  defp new_cell_info() do
+    %{
+      revision: 0,
+      deltas: [],
+      status: :fresh,
+      evaluated_at: nil
+    }
+  end
+
+  defp set!(data, changes) do
+    Enum.reduce(changes, data, fn {key, value}, info ->
+      Map.replace!(info, key, value)
+    end)
+  end
+
+  defp set_cell_info!(data, cell_id, changes) do
+    update_cell_info!(data, cell_id, fn info ->
+      Enum.reduce(changes, info, fn {key, value}, info ->
+        Map.replace!(info, key, value)
+      end)
+    end)
+  end
+
+  defp update_cell_info!(data, cell_id, fun) do
+    cell_infos = Map.update!(data.cell_infos, cell_id, fun)
+    set!(data, cell_infos: cell_infos)
+  end
+
+  defp set_section_info!(data, section_id, changes) do
+    update_section_info!(data, section_id, fn info ->
+      Enum.reduce(changes, info, fn {key, value}, info ->
+        Map.replace!(info, key, value)
+      end)
+    end)
+  end
+
+  defp update_section_info!(data, section_id, fun) do
+    section_infos = Map.update!(data.section_infos, section_id, fun)
+    set!(data, section_infos: section_infos)
+  end
+
+  defp reduce(data, list, reducer) do
+    Enum.reduce(list, data, fn elem, data -> reducer.(data, elem) end)
+  end
+
+  @doc """
+  Finds the cell that's currently being evaluated in the given section.
+  """
+  @spec get_evaluating_cell_id(t(), Section.id()) :: Cell.id() | nil
+  def get_evaluating_cell_id(data, section_id) do
+    info = data.section_infos[section_id]
+    info && info.evaluating_cell_id
+  end
+end
diff --git a/lib/live_book/session_supervisor.ex b/lib/live_book/session_supervisor.ex
index f9b5b9e3f..accb8fcc6 100644
--- a/lib/live_book/session_supervisor.ex
+++ b/lib/live_book/session_supervisor.ex
@@ -26,7 +26,7 @@ defmodule LiveBook.SessionSupervisor do
 
   Broadcasts `{:session_created, id}` message under the `"sessions"` topic.
   """
-  @spec create_session() :: {:ok, Session.session_id()} | {:error, any()}
+  @spec create_session() :: {:ok, Section.id()} | {:error, any()}
   def create_session() do
     id = Utils.random_id()
 
@@ -52,7 +52,7 @@ defmodule LiveBook.SessionSupervisor do
 
   Broadcasts `{:session_delete, id}` message under the `"sessions"` topic.
   """
-  @spec delete_session(Session.session_id()) :: :ok
+  @spec delete_session(Section.id()) :: :ok
   def delete_session(id) do
     Session.stop(id)
     broadcast_sessions_message({:session_deleted, id})
@@ -66,7 +66,7 @@ defmodule LiveBook.SessionSupervisor do
   @doc """
   Returns ids of all the running session processes.
   """
-  @spec get_session_ids() :: list(Session.session_id())
+  @spec get_session_ids() :: list(Section.id())
   def get_session_ids() do
     :global.registered_names()
     |> Enum.flat_map(fn
@@ -78,7 +78,7 @@ defmodule LiveBook.SessionSupervisor do
   @doc """
   Checks if a session process with the given id exists.
   """
-  @spec session_exists?(Session.session_id()) :: boolean()
+  @spec session_exists?(Section.id()) :: boolean()
   def session_exists?(id) do
     :global.whereis_name({:session, id}) != :undefined
   end
@@ -86,7 +86,7 @@ defmodule LiveBook.SessionSupervisor do
   @doc """
   Retrieves pid of a session process identified by the given id.
   """
-  @spec get_session_pid(Session.session_id()) :: {:ok, pid()} | {:error, :nonexistent}
+  @spec get_session_pid(Section.id()) :: {:ok, pid()} | {:error, :nonexistent}
   def get_session_pid(id) do
     case :global.whereis_name({:session, id}) do
       :undefined -> {:error, :nonexistent}
diff --git a/test/live_book/evaluator_test.exs b/test/live_book/evaluator_test.exs
index bcd4ad712..dc5dc5f80 100644
--- a/test/live_book/evaluator_test.exs
+++ b/test/live_book/evaluator_test.exs
@@ -16,44 +16,47 @@ defmodule LiveBook.EvaluatorTest do
       x + y
       """
 
-      result = Evaluator.evaluate_code(evaluator, code, 1)
+      Evaluator.evaluate_code(evaluator, self(), code, :code_1)
 
-      assert result == {:ok, 3}
+      assert_receive {:evaluator_response, :code_1, {:ok, 3}}
     end
 
     test "given no prev_ref does not see previous evaluation context", %{evaluator: evaluator} do
-      Evaluator.evaluate_code(evaluator, "x = 1", :code_1)
+      Evaluator.evaluate_code(evaluator, self(), "x = 1", :code_1)
+      assert_receive {:evaluator_response, :code_1, _}
 
-      result = Evaluator.evaluate_code(evaluator, "x", :code_2)
+      Evaluator.evaluate_code(evaluator, self(), "x", :code_2)
 
-      assert {:error, _kind, %CompileError{description: "undefined function x/0"}, _stacktrace} =
-               result
+      assert_receive {:evaluator_response, :code_2,
+                      {:error, _kind, %CompileError{description: "undefined function x/0"},
+                       _stacktrace}}
     end
 
     test "given prev_ref sees previous evaluation context", %{evaluator: evaluator} do
-      Evaluator.evaluate_code(evaluator, "x = 1", :code_1)
+      Evaluator.evaluate_code(evaluator, self(), "x = 1", :code_1)
+      assert_receive {:evaluator_response, :code_1, _}
 
-      result = Evaluator.evaluate_code(evaluator, "x", :code_2, :code_1)
+      Evaluator.evaluate_code(evaluator, self(), "x", :code_2, :code_1)
 
-      assert result == {:ok, 1}
+      assert_receive {:evaluator_response, :code_2, {:ok, 1}}
     end
 
-    test "given invalid prev_ref raises an error", %{evaluator: evaluator} do
-      assert_raise ArgumentError, fn ->
-        Evaluator.evaluate_code(evaluator, ":ok", :code_1, :code_nonexistent)
-      end
+    test "given invalid prev_ref just uses default context", %{evaluator: evaluator} do
+      Evaluator.evaluate_code(evaluator, self(), ":hey", :code_1, :code_nonexistent)
+
+      assert_receive {:evaluator_response, :code_1, {:ok, :hey}}
     end
 
     test "captures standard output and sends it to the caller", %{evaluator: evaluator} do
-      Evaluator.evaluate_code(evaluator, ~s{IO.puts("hey")}, :code_1)
+      Evaluator.evaluate_code(evaluator, self(), ~s{IO.puts("hey")}, :code_1)
 
-      assert_received {:evaluator_stdout, :code_1, "hey\n"}
+      assert_receive {:evaluator_stdout, :code_1, "hey\n"}
     end
 
     test "using standard input results in an immediate error", %{evaluator: evaluator} do
-      result = Evaluator.evaluate_code(evaluator, ~s{IO.gets("> ")}, :code_1)
+      Evaluator.evaluate_code(evaluator, self(), ~s{IO.gets("> ")}, :code_1)
 
-      assert result == {:ok, {:error, :enotsup}}
+      assert_receive {:evaluator_response, :code_1, {:ok, {:error, :enotsup}}}
     end
 
     test "returns error along with its kind and stacktrace", %{evaluator: evaluator} do
@@ -61,9 +64,10 @@ defmodule LiveBook.EvaluatorTest do
       List.first(%{})
       """
 
-      result = Evaluator.evaluate_code(evaluator, code, :code_1)
+      Evaluator.evaluate_code(evaluator, self(), code, :code_1)
 
-      assert {:error, :error, %FunctionClauseError{}, [{List, :first, 1, _location}]} = result
+      assert_receive {:evaluator_response, :code_1,
+                      {:error, :error, %FunctionClauseError{}, [{List, :first, 1, _location}]}}
     end
 
     test "in case of an error returns only the relevant part of stacktrace", %{
@@ -87,25 +91,28 @@ defmodule LiveBook.EvaluatorTest do
       Cat.meow()
       """
 
-      result = Evaluator.evaluate_code(evaluator, code, :code_1)
+      Evaluator.evaluate_code(evaluator, self(), code, :code_1)
 
       expected_stacktrace = [
         {Math, :bad_math, 0, [file: 'nofile', line: 3]},
         {Cat, :meow, 0, [file: 'nofile', line: 10]}
       ]
 
-      assert {:error, _kind, _error, ^expected_stacktrace} = result
+      assert_receive {:evaluator_response, :code_1, {:error, _kind, _error, ^expected_stacktrace}}
     end
   end
 
   describe "forget_evaluation/2" do
     test "invalidates the given reference", %{evaluator: evaluator} do
-      Evaluator.evaluate_code(evaluator, "x = 1", :code_1)
-      Evaluator.forget_evaluation(evaluator, :code_1)
+      Evaluator.evaluate_code(evaluator, self(), "x = 1", :code_1)
+      assert_receive {:evaluator_response, :code_1, _}
 
-      assert_raise ArgumentError, fn ->
-        Evaluator.evaluate_code(evaluator, ":ok", :code_2, :code_1)
-      end
+      Evaluator.forget_evaluation(evaluator, :code_1)
+      Evaluator.evaluate_code(evaluator, self(), "x", :code_2, :code_1)
+
+      assert_receive {:evaluator_response, :code_2,
+                      {:error, _kind, %CompileError{description: "undefined function x/0"},
+                       _stacktrace}}
     end
   end
 end
diff --git a/test/live_book/session/data_test.exs b/test/live_book/session/data_test.exs
new file mode 100644
index 000000000..0e6693d49
--- /dev/null
+++ b/test/live_book/session/data_test.exs
@@ -0,0 +1,335 @@
+defmodule LiveBook.Session.DataTest do
+  use ExUnit.Case, async: true
+
+  alias LiveBook.Session.Data
+
+  describe "apply_operation/2 given :insert_section" do
+    test "adds new section to notebook and session info" do
+      data = Data.new()
+
+      operation = {:insert_section, 0, "s1"}
+
+      assert {:ok,
+              %{
+                notebook: %{
+                  sections: [%{id: "s1"}]
+                },
+                section_infos: %{"s1" => _}
+              }} = Data.apply_operation(data, operation)
+    end
+  end
+
+  describe "apply_operation/2 given :insert_cell" do
+    test "returns an error given invalid section id" do
+      data = Data.new()
+      operation = {:insert_cell, "nonexistent", 0, :elixir, "c1"}
+      assert :error = Data.apply_operation(data, operation)
+    end
+
+    test "insert_cell adds new cell to notebook and session info" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"}
+        ])
+
+      operation = {:insert_cell, "s1", 0, :elixir, "c1"}
+
+      assert {:ok,
+              %{
+                notebook: %{
+                  sections: [
+                    %{cells: [%{id: "c1"}]}
+                  ]
+                },
+                cell_infos: %{"c1" => _}
+              }} = Data.apply_operation(data, operation)
+    end
+  end
+
+  describe "apply_operation/2 given :delete_section" do
+    test "returns an error given invalid section id" do
+      data = Data.new()
+      operation = {:delete_section, "nonexistent"}
+      assert :error = Data.apply_operation(data, operation)
+    end
+
+    test "returns an error for an evaluating section" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"},
+          {:queue_cell_evaluation, "c1"}
+        ])
+
+      operation = {:delete_section, "s1"}
+      assert :error = Data.apply_operation(data, operation)
+    end
+
+    test "removes the section from notebook and session info, adds to deleted sections" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"}
+        ])
+
+      operation = {:delete_section, "s1"}
+      empty_map = %{}
+
+      assert {:ok,
+              %{
+                notebook: %{
+                  sections: []
+                },
+                section_infos: ^empty_map,
+                deleted_sections: [%{id: "s1"}]
+              }} = Data.apply_operation(data, operation)
+    end
+  end
+
+  describe "apply_operation/2 given :delete_cell" do
+    test "returns an error given invalid cell id" do
+      data = Data.new()
+      operation = {:delete_cell, "nonexistent"}
+      assert :error = Data.apply_operation(data, operation)
+    end
+
+    test "returns an error for an evaluating cell" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"},
+          {:queue_cell_evaluation, "c1"}
+        ])
+
+      operation = {:delete_cell, "c1"}
+      assert :error = Data.apply_operation(data, operation)
+    end
+
+    test "removes the cell from notebook and session info, adds to deleted cells" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"}
+        ])
+
+      operation = {:delete_cell, "c1"}
+      empty_map = %{}
+
+      assert {:ok,
+              %{
+                notebook: %{
+                  sections: [%{cells: []}]
+                },
+                cell_infos: ^empty_map,
+                deleted_cells: [%{id: "c1"}]
+              }} = Data.apply_operation(data, operation)
+    end
+
+    test "unqueues the cell if it's queued for evaluation" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"},
+          {:insert_cell, "s1", 1, :elixir, "c2"},
+          {:queue_cell_evaluation, "c1"},
+          {:queue_cell_evaluation, "c2"}
+        ])
+
+      operation = {:delete_cell, "c2"}
+
+      assert {:ok,
+              %{
+                section_infos: %{"s1" => %{evaluation_queue: []}}
+              }} = Data.apply_operation(data, operation)
+    end
+
+    test "marks evaluated child cells as stale" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"},
+          {:insert_cell, "s1", 1, :elixir, "c2"},
+          # Evaluate both cells
+          {:queue_cell_evaluation, "c1"},
+          {:add_cell_evaluation_response, "c1", {:ok, [1, 2, 3]}},
+          {:queue_cell_evaluation, "c2"},
+          {:add_cell_evaluation_response, "c2", {:ok, [1, 2, 3]}}
+        ])
+
+      operation = {:delete_cell, "c1"}
+
+      assert {:ok,
+              %{
+                cell_infos: %{"c2" => %{status: :stale}}
+              }} = Data.apply_operation(data, operation)
+    end
+  end
+
+  describe "apply_operation/2 given :queue_cell_evaluation" do
+    test "returns an error given invalid cell id" do
+      data = Data.new()
+      operation = {:queue_cell_evaluation, "nonexistent"}
+      assert :error = Data.apply_operation(data, operation)
+    end
+
+    test "returns an error given non-elixir cell" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :markdown, "c1"}
+        ])
+
+      operation = {:queue_cell_evaluation, "c1"}
+      assert :error = Data.apply_operation(data, operation)
+    end
+
+    test "returns an error for an evaluating cell" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"},
+          {:queue_cell_evaluation, "c1"}
+        ])
+
+      operation = {:queue_cell_evaluation, "c1"}
+      assert :error = Data.apply_operation(data, operation)
+    end
+
+    test "marks the cell as evaluating if the corresponding section is idle" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"}
+        ])
+
+      operation = {:queue_cell_evaluation, "c1"}
+
+      assert {:ok,
+              %{
+                cell_infos: %{"c1" => %{status: :evaluating}},
+                section_infos: %{"s1" => %{evaluating_cell_id: "c1", evaluation_queue: []}}
+              }} = Data.apply_operation(data, operation)
+    end
+
+    test "marks the cell as queued if the corresponding section is already evaluating" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"},
+          {:insert_cell, "s1", 1, :elixir, "c2"},
+          {:queue_cell_evaluation, "c1"}
+        ])
+
+      operation = {:queue_cell_evaluation, "c2"}
+
+      assert {:ok,
+              %{
+                cell_infos: %{"c2" => %{status: :queued}},
+                section_infos: %{"s1" => %{evaluating_cell_id: "c1", evaluation_queue: ["c2"]}}
+              }} = Data.apply_operation(data, operation)
+    end
+  end
+
+  describe "apply_operation/2 given :add_cell_evaluation_stdout" do
+    test "update the cell output" do
+      # TODO assert against output being updated once we do so
+    end
+  end
+
+  describe "apply_operation/2 given :add_cell_evaluation_response" do
+    test "update the cell output" do
+      # TODO assert against output being updated once we do so
+    end
+
+    test "marks the cell as evaluated" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"},
+          {:insert_cell, "s1", 1, :elixir, "c2"},
+          {:queue_cell_evaluation, "c1"}
+        ])
+
+      operation = {:add_cell_evaluation_response, "c1", {:ok, [1, 2, 3]}}
+
+      assert {:ok,
+              %{
+                cell_infos: %{"c1" => %{status: :evaluated}},
+                section_infos: %{"s1" => %{evaluating_cell_id: nil, evaluation_queue: []}}
+              }} = Data.apply_operation(data, operation)
+    end
+
+    test "marks next queued cell in this section as evaluating if there is one" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"},
+          {:insert_cell, "s1", 1, :elixir, "c2"},
+          {:queue_cell_evaluation, "c1"},
+          {:queue_cell_evaluation, "c2"}
+        ])
+
+      operation = {:add_cell_evaluation_response, "c1", {:ok, [1, 2, 3]}}
+
+      assert {:ok,
+              %{
+                cell_infos: %{"c2" => %{status: :evaluating}},
+                section_infos: %{"s1" => %{evaluating_cell_id: "c2", evaluation_queue: []}}
+              }} = Data.apply_operation(data, operation)
+    end
+
+    test "if parent cells are not executed, marks them for evaluation first" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"},
+          {:insert_cell, "s1", 1, :elixir, "c2"}
+        ])
+
+      operation = {:queue_cell_evaluation, "c2"}
+
+      assert {:ok,
+              %{
+                cell_infos: %{
+                  "c1" => %{status: :evaluating},
+                  "c2" => %{status: :queued}
+                },
+                section_infos: %{"s1" => %{evaluating_cell_id: "c1", evaluation_queue: ["c2"]}}
+              }} = Data.apply_operation(data, operation)
+    end
+
+    test "marks evaluated child cells as stale" do
+      data =
+        data_after_operations!([
+          {:insert_section, 0, "s1"},
+          {:insert_cell, "s1", 0, :elixir, "c1"},
+          {:insert_cell, "s1", 1, :elixir, "c2"},
+          # Evaluate both cells
+          {:queue_cell_evaluation, "c1"},
+          {:add_cell_evaluation_response, "c1", {:ok, [1, 2, 3]}},
+          {:queue_cell_evaluation, "c2"},
+          {:add_cell_evaluation_response, "c2", {:ok, [1, 2, 3]}},
+          # Queue the first cell again
+          {:queue_cell_evaluation, "c1"}
+        ])
+
+      operation = {:add_cell_evaluation_response, "c1", {:ok, [1, 2, 3]}}
+
+      assert {:ok,
+              %{
+                cell_infos: %{"c2" => %{status: :stale}}
+              }} = Data.apply_operation(data, operation)
+    end
+  end
+
+  defp data_after_operations!(operations) do
+    Enum.reduce(operations, Data.new(), fn operation, data ->
+      case Data.apply_operation(data, operation) do
+        {:ok, data} ->
+          data
+
+        :error ->
+          raise "failed to set up test data, operation #{inspect(operation)} returned an error"
+      end
+    end)
+  end
+end
diff --git a/test/live_book/session_test.exs b/test/live_book/session_test.exs
new file mode 100644
index 000000000..5bac527d1
--- /dev/null
+++ b/test/live_book/session_test.exs
@@ -0,0 +1,83 @@
+defmodule LiveBook.SessionTest do
+  use ExUnit.Case, async: true
+
+  alias LiveBook.Session
+
+  setup do
+    {:ok, _} = Session.start_link("1")
+    %{session_id: "1"}
+  end
+
+  describe "insert_section/2" do
+    test "sends an insert opreation to subscribers", %{session_id: session_id} do
+      Phoenix.PubSub.subscribe(LiveBook.PubSub, "sessions:#{session_id}")
+
+      Session.insert_section(session_id, 0)
+      assert_receive {:operation, {:insert_section, 0, _id}}
+    end
+  end
+
+  describe "insert_cell/4" do
+    test "sends an insert opreation to subscribers", %{session_id: session_id} do
+      Phoenix.PubSub.subscribe(LiveBook.PubSub, "sessions:#{session_id}")
+
+      Session.insert_section(session_id, 0)
+      assert_receive {:operation, {:insert_section, 0, section_id}}
+
+      Session.insert_cell(session_id, section_id, 0, :elixir)
+      assert_receive {:operation, {:insert_cell, ^section_id, 0, :elixir, _id}}
+    end
+  end
+
+  describe "delete_section/2" do
+    test "sends a delete opreation to subscribers", %{session_id: session_id} do
+      Phoenix.PubSub.subscribe(LiveBook.PubSub, "sessions:#{session_id}")
+
+      {section_id, _cell_id} = insert_section_and_cell(session_id)
+
+      Session.delete_section(session_id, section_id)
+      assert_receive {:operation, {:delete_section, ^section_id}}
+    end
+  end
+
+  describe "delete_cell/2" do
+    test "sends a delete opreation to subscribers", %{session_id: session_id} do
+      Phoenix.PubSub.subscribe(LiveBook.PubSub, "sessions:#{session_id}")
+
+      {_section_id, cell_id} = insert_section_and_cell(session_id)
+
+      Session.delete_cell(session_id, cell_id)
+      assert_receive {:operation, {:delete_cell, ^cell_id}}
+    end
+  end
+
+  describe "queue_cell_evaluation/2" do
+    test "sends a queue evaluation operation to subscribers", %{session_id: session_id} do
+      Phoenix.PubSub.subscribe(LiveBook.PubSub, "sessions:#{session_id}")
+
+      {_section_id, cell_id} = insert_section_and_cell(session_id)
+
+      Session.queue_cell_evaluation(session_id, cell_id)
+      assert_receive {:operation, {:queue_cell_evaluation, ^cell_id}}
+    end
+
+    test "triggers evaluation and sends update operation once it finishes",
+         %{session_id: session_id} do
+      Phoenix.PubSub.subscribe(LiveBook.PubSub, "sessions:#{session_id}")
+
+      {_section_id, cell_id} = insert_section_and_cell(session_id)
+
+      Session.queue_cell_evaluation(session_id, cell_id)
+      assert_receive {:operation, {:add_cell_evaluation_response, ^cell_id, _}}
+    end
+  end
+
+  defp insert_section_and_cell(session_id) do
+    Session.insert_section(session_id, 0)
+    assert_receive {:operation, {:insert_section, 0, section_id}}
+    Session.insert_cell(session_id, section_id, 0, :elixir)
+    assert_receive {:operation, {:insert_cell, ^section_id, 0, :elixir, cell_id}}
+
+    {section_id, cell_id}
+  end
+end
diff --git a/test/test_helper.exs b/test/test_helper.exs
index 869559e70..7eb55657b 100644
--- a/test/test_helper.exs
+++ b/test/test_helper.exs
@@ -1 +1 @@
-ExUnit.start()
+ExUnit.start(assert_receive_timeout: 500)