diff --git a/lib/livebook/notebook.ex b/lib/livebook/notebook.ex
index d95082adc..333e13eae 100644
--- a/lib/livebook/notebook.ex
+++ b/lib/livebook/notebook.ex
@@ -182,38 +182,39 @@ defmodule Livebook.Notebook do
   end
 
   @doc """
-  Returns a list of Elixir cells that the given cell depends on.
-
-  The cells are ordered starting from the most direct parent.
+  Returns a list of `{cell, section}` pairs including all Elixir cells in order.
   """
-  @spec parent_cells(t(), Cell.id()) :: list(Cell.t())
-  def parent_cells(notebook, cell_id) do
-    with {:ok, _, section} <- fetch_cell_and_section(notebook, cell_id) do
-      # A cell depends on all previous cells within the same section.
-      section.cells
-      |> Enum.take_while(&(&1.id != cell_id))
-      |> Enum.reverse()
-      |> Enum.filter(&(&1.type == :elixir))
-    else
-      _ -> []
-    end
+  @spec elixir_cells_with_section(t()) :: list({Cell.t(), Section.t()})
+  def elixir_cells_with_section(notebook) do
+    for section <- notebook.sections,
+        cell <- section.cells,
+        cell.type == :elixir,
+        do: {cell, section}
   end
 
   @doc """
-  Returns a list of Elixir cells that depend on the given cell.
+  Returns a list of Elixir cells (each with section) that the given cell depends on.
+
+  The cells are ordered starting from the most direct parent.
+  """
+  @spec parent_cells_with_section(t(), Cell.id()) :: list({Cell.t(), Section.t()})
+  def parent_cells_with_section(notebook, cell_id) do
+    notebook
+    |> elixir_cells_with_section()
+    |> Enum.take_while(fn {cell, _} -> cell.id != cell_id end)
+    |> Enum.reverse()
+  end
+
+  @doc """
+  Returns a list of Elixir cells (each with section) 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} <- fetch_cell_and_section(notebook, cell_id) do
-      # A cell affects all the cells below it within the same section.
-      section.cells
-      |> Enum.drop_while(&(&1.id != cell_id))
-      |> Enum.drop(1)
-      |> Enum.filter(&(&1.type == :elixir))
-    else
-      _ -> []
-    end
+  @spec child_cells_with_section(t(), Cell.id()) :: list({Cell.t(), Section.t()})
+  def child_cells_with_section(notebook, cell_id) do
+    notebook
+    |> elixir_cells_with_section()
+    |> Enum.drop_while(fn {cell, _} -> cell.id != cell_id end)
+    |> Enum.drop(1)
   end
 end
diff --git a/lib/livebook/session.ex b/lib/livebook/session.ex
index 585c5fce1..e950abe3f 100644
--- a/lib/livebook/session.ex
+++ b/lib/livebook/session.ex
@@ -505,17 +505,17 @@ defmodule Livebook.Session do
     start_evaluation(state, cell, section)
   end
 
-  defp handle_action(state, {:stop_evaluation, section}) do
+  defp handle_action(state, {:stop_evaluation, _section}) do
     if state.data.runtime do
-      Runtime.drop_container(state.data.runtime, section.id)
+      Runtime.drop_container(state.data.runtime, :main)
     end
 
     state
   end
 
-  defp handle_action(state, {:forget_evaluation, cell, section}) do
+  defp handle_action(state, {:forget_evaluation, cell, _section}) do
     if state.data.runtime do
-      Runtime.forget_evaluation(state.data.runtime, section.id, cell.id)
+      Runtime.forget_evaluation(state.data.runtime, :main, cell.id)
     end
 
     state
@@ -539,14 +539,14 @@ defmodule Livebook.Session do
     Phoenix.PubSub.broadcast(Livebook.PubSub, "sessions:#{session_id}", message)
   end
 
-  defp start_evaluation(state, cell, section) do
+  defp start_evaluation(state, cell, _section) do
     prev_ref =
-      case Notebook.parent_cells(state.data.notebook, cell.id) do
-        [parent | _] -> parent.id
+      case Notebook.parent_cells_with_section(state.data.notebook, cell.id) do
+        [{parent, _} | _] -> parent.id
         [] -> :initial
       end
 
-    Runtime.evaluate_code(state.data.runtime, cell.source, section.id, cell.id, prev_ref)
+    Runtime.evaluate_code(state.data.runtime, cell.source, :main, cell.id, prev_ref)
 
     state
   end
diff --git a/lib/livebook/session/data.ex b/lib/livebook/session/data.ex
index b009d0e26..9e14fe627 100644
--- a/lib/livebook/session/data.ex
+++ b/lib/livebook/session/data.ex
@@ -192,14 +192,14 @@ defmodule Livebook.Session.Data do
         :evaluating ->
           data
           |> with_actions()
-          |> clear_section_evaluation(section)
+          |> clear_evaluation()
           |> add_action({:stop_evaluation, section})
 
         :queued ->
           data
           |> with_actions()
           |> unqueue_cell_evaluation(cell, section)
-          |> unqueue_dependent_cells_evaluation(cell, section)
+          |> unqueue_dependent_cells_evaluation(cell)
           |> mark_dependent_cells_as_stale(cell)
 
         _ ->
@@ -233,7 +233,7 @@ defmodule Livebook.Session.Data do
          :ready <- data.cell_infos[cell.id].evaluation_status do
       data
       |> with_actions()
-      |> queue_prerequisite_cells_evaluation(cell, section)
+      |> queue_prerequisite_cells_evaluation(cell)
       |> queue_cell_evaluation(cell, section)
       |> maybe_evaluate_queued()
       |> wrap_ok()
@@ -275,7 +275,7 @@ defmodule Livebook.Session.Data do
         :evaluating ->
           data
           |> with_actions()
-          |> clear_section_evaluation(section)
+          |> clear_evaluation()
           |> add_action({:stop_evaluation, section})
           |> wrap_ok()
 
@@ -283,7 +283,7 @@ defmodule Livebook.Session.Data do
           data
           |> with_actions()
           |> unqueue_cell_evaluation(cell, section)
-          |> unqueue_dependent_cells_evaluation(cell, section)
+          |> unqueue_dependent_cells_evaluation(cell)
           |> mark_dependent_cells_as_stale(cell)
           |> wrap_ok()
 
@@ -378,7 +378,7 @@ defmodule Livebook.Session.Data do
     data
     |> with_actions()
     |> set!(runtime: runtime)
-    |> reduce(data.notebook.sections, &clear_section_evaluation/2)
+    |> clear_evaluation()
     |> wrap_ok()
   end
 
@@ -448,30 +448,22 @@ defmodule Livebook.Session.Data do
     new_idx = (idx + offset) |> clamp_index(section.cells)
 
     updated_notebook = Notebook.move_cell(data.notebook, section.id, idx, new_idx)
-    {:ok, updated_section} = Notebook.fetch_section(updated_notebook, section.id)
 
-    elixir_cell_ids_before = elixir_cell_ids(section.cells)
-    elixir_cell_ids_after = elixir_cell_ids(updated_section.cells)
+    cells_with_section_before = Notebook.elixir_cells_with_section(data.notebook)
+    cells_with_section_after = Notebook.elixir_cells_with_section(updated_notebook)
 
-    # If the order of Elixir cells stays the same, no need to invalidate anything
-    affected_cells =
-      if elixir_cell_ids_before != elixir_cell_ids_after do
-        affected_from_idx = min(idx, new_idx)
-        Enum.slice(updated_section.cells, affected_from_idx..-1)
-      else
-        []
-      end
+    affected_cells_with_section =
+      cells_with_section_after
+      |> Enum.zip(cells_with_section_before)
+      |> Enum.drop_while(fn {{cell_before, _}, {cell_after, _}} ->
+        cell_before.id == cell_after.id
+      end)
+      |> Enum.map(fn {new, _old} -> new end)
 
     data_actions
     |> set!(notebook: updated_notebook)
-    |> mark_cells_as_stale(affected_cells)
-    |> unqueue_cells_evaluation(affected_cells, section)
-  end
-
-  defp elixir_cell_ids(cells) do
-    cells
-    |> Enum.filter(&(&1.type == :elixir))
-    |> Enum.map(& &1.id)
+    |> mark_cells_as_stale(affected_cells_with_section)
+    |> unqueue_cells_evaluation(affected_cells_with_section)
   end
 
   defp clamp_index(index, list) do
@@ -534,13 +526,15 @@ defmodule Livebook.Session.Data do
   end
 
   defp mark_dependent_cells_as_stale({data, _} = data_actions, cell) do
-    child_cells = Notebook.child_cells(data.notebook, cell.id)
+    child_cells = Notebook.child_cells_with_section(data.notebook, cell.id)
     mark_cells_as_stale(data_actions, child_cells)
   end
 
-  defp mark_cells_as_stale({data, _} = data_actions, cells) do
+  defp mark_cells_as_stale({data, _} = data_actions, cells_with_section) do
     invalidated_cells =
-      Enum.filter(cells, fn cell ->
+      cells_with_section
+      |> Enum.map(fn {cell, _section} -> cell end)
+      |> Enum.filter(fn cell ->
         cell.type == :elixir and data.cell_infos[cell.id].validity_status == :evaluated
       end)
 
@@ -548,26 +542,39 @@ defmodule Livebook.Session.Data do
     |> reduce(invalidated_cells, &set_cell_info!(&1, &2.id, validity_status: :stale))
   end
 
-  # If there are idle sections with non-empty evaluation queue,
-  # the next queued cell for evaluation.
   defp maybe_evaluate_queued({data, _} = data_actions) do
-    Enum.reduce(data.notebook.sections, data_actions, fn section, data_actions ->
-      {data, _} = data_actions
+    ongoing_evaluation? =
+      Enum.any?(data.notebook.sections, fn section ->
+        data.section_infos[section.id].evaluating_cell_id != nil
+      end)
 
-      case data.section_infos[section.id] do
-        %{evaluating_cell_id: nil, evaluation_queue: [id | ids]} ->
-          cell = Enum.find(section.cells, &(&1.id == id))
+    if ongoing_evaluation? do
+      # A section is evaluating, so we don't start any new evaluation
+      data_actions
+    else
+      Enum.find_value(data.notebook.sections, data_actions, fn section ->
+        case data.section_infos[section.id] do
+          %{evaluating_cell_id: nil, evaluation_queue: [id | ids]} ->
+            # The section is idle and has cells queued for evaluation, so let's start the evaluation
+            cell = Enum.find(section.cells, &(&1.id == id))
 
-          data_actions
-          |> set!(notebook: Notebook.update_cell(data.notebook, id, &%{&1 | outputs: []}))
-          |> set_cell_info!(id, evaluation_status: :evaluating)
-          |> set_section_info!(section.id, evaluating_cell_id: id, evaluation_queue: ids)
-          |> add_action({:start_evaluation, cell, section})
+            data_actions
+            |> set!(notebook: Notebook.update_cell(data.notebook, id, &%{&1 | outputs: []}))
+            |> set_cell_info!(id, evaluation_status: :evaluating)
+            |> set_section_info!(section.id, evaluating_cell_id: id, evaluation_queue: ids)
+            |> add_action({:start_evaluation, cell, section})
 
-        _ ->
-          data_actions
-      end
-    end)
+          _ ->
+            # The section is neither evaluating nor queued, so let's check the next section
+            nil
+        end
+      end)
+    end
+  end
+
+  defp clear_evaluation({data, _} = data_actions) do
+    data_actions
+    |> reduce(data.notebook.sections, &clear_section_evaluation/2)
   end
 
   defp clear_section_evaluation(data_actions, section) do
@@ -579,31 +586,37 @@ defmodule Livebook.Session.Data do
     )
   end
 
-  defp queue_prerequisite_cells_evaluation({data, _} = data_actions, cell, section) do
+  defp queue_prerequisite_cells_evaluation({data, _} = data_actions, cell) do
     prerequisites_queue =
       data.notebook
-      |> Notebook.parent_cells(cell.id)
-      |> Enum.take_while(fn parent ->
-        info = data.cell_infos[parent.id]
-        info.validity_status == :fresh and info.evaluation_status == :ready
+      |> Notebook.parent_cells_with_section(cell.id)
+      |> Enum.take_while(fn {parent_cell, _section} ->
+        info = data.cell_infos[parent_cell.id]
+        info.validity_status != :evaluated and info.evaluation_status == :ready
       end)
       |> Enum.reverse()
 
     data_actions
-    |> reduce(prerequisites_queue, &queue_cell_evaluation(&1, &2, section))
+    |> reduce(prerequisites_queue, fn data_actions, {cell, section} ->
+      queue_cell_evaluation(data_actions, cell, section)
+    end)
   end
 
-  defp unqueue_dependent_cells_evaluation({data, _} = data_actions, cell, section) do
-    dependent_cells = Notebook.child_cells(data.notebook, cell.id)
-    unqueue_cells_evaluation(data_actions, dependent_cells, section)
+  defp unqueue_dependent_cells_evaluation({data, _} = data_actions, cell) do
+    dependent_cells = Notebook.child_cells_with_section(data.notebook, cell.id)
+    unqueue_cells_evaluation(data_actions, dependent_cells)
   end
 
-  defp unqueue_cells_evaluation({data, _} = data_actions, cells, section) do
-    queued_cells =
-      Enum.filter(cells, fn cell -> data.cell_infos[cell.id].evaluation_status == :queued end)
+  defp unqueue_cells_evaluation({data, _} = data_actions, cells_with_section) do
+    queued_cells_with_section =
+      Enum.filter(cells_with_section, fn {cell, _} ->
+        data.cell_infos[cell.id].evaluation_status == :queued
+      end)
 
     data_actions
-    |> reduce(queued_cells, &unqueue_cell_evaluation(&1, &2, section))
+    |> reduce(queued_cells_with_section, fn data_actions, {cell, section} ->
+      unqueue_cell_evaluation(data_actions, cell, section)
+    end)
   end
 
   defp set_notebook_name({data, _} = data_actions, name) do
diff --git a/lib/livebook_web/live/session_live.ex b/lib/livebook_web/live/session_live.ex
index 9662ce0d2..cffb80690 100644
--- a/lib/livebook_web/live/session_live.ex
+++ b/lib/livebook_web/live/session_live.ex
@@ -374,9 +374,7 @@ defmodule LivebookWeb.SessionLive do
           socket.assigns.focused_cell_id
         )
 
-      cells = Notebook.child_cells(socket.assigns.data.notebook, cell.id)
-
-      for cell <- cells, cell.type == :elixir do
+      for {cell, _} <- Notebook.child_cells_with_section(socket.assigns.data.notebook, cell.id) do
         Session.queue_cell_evaluation(socket.assigns.session_id, cell.id)
       end
     end
diff --git a/test/livebook/session/data_test.exs b/test/livebook/session/data_test.exs
index b460de146..a53f544f0 100644
--- a/test/livebook/session/data_test.exs
+++ b/test/livebook/session/data_test.exs
@@ -309,6 +309,31 @@ defmodule Livebook.Session.DataTest do
               }, []} = Data.apply_operation(data, operation)
     end
 
+    test "marks relevant cells in further sections as stale" do
+      data =
+        data_after_operations!([
+          {:insert_section, self(), 0, "s1"},
+          {:insert_cell, self(), "s1", 0, :elixir, "c1"},
+          {:insert_cell, self(), "s1", 1, :elixir, "c2"},
+          {:insert_section, self(), 1, "s2"},
+          {:insert_cell, self(), "s2", 0, :elixir, "c3"},
+          # Evaluate cells
+          {:queue_cell_evaluation, self(), "c1"},
+          {:add_cell_evaluation_response, self(), "c1", {:ok, nil}},
+          {:queue_cell_evaluation, self(), "c2"},
+          {:add_cell_evaluation_response, self(), "c2", {:ok, nil}},
+          {:queue_cell_evaluation, self(), "c3"},
+          {:add_cell_evaluation_response, self(), "c3", {:ok, nil}}
+        ])
+
+      operation = {:move_cell, self(), "c1", 1}
+
+      assert {:ok,
+              %{
+                cell_infos: %{"c3" => %{validity_status: :stale}}
+              }, []} = Data.apply_operation(data, operation)
+    end
+
     test "moving a markdown cell does not change validity" do
       data =
         data_after_operations!([
@@ -456,6 +481,75 @@ defmodule Livebook.Session.DataTest do
                 section_infos: %{"s1" => %{evaluating_cell_id: "c1", evaluation_queue: ["c2"]}}
               }, []} = Data.apply_operation(data, operation)
     end
+
+    test "marks the cell as queued if a previous section is already evaluating" do
+      data =
+        data_after_operations!([
+          {:insert_section, self(), 0, "s1"},
+          {:insert_cell, self(), "s1", 0, :elixir, "c1"},
+          {:insert_section, self(), 1, "s2"},
+          {:insert_cell, self(), "s2", 0, :elixir, "c2"},
+          {:queue_cell_evaluation, self(), "c1"}
+        ])
+
+      operation = {:queue_cell_evaluation, self(), "c2"}
+
+      assert {:ok,
+              %{
+                cell_infos: %{"c2" => %{evaluation_status: :queued}},
+                section_infos: %{
+                  "s1" => %{evaluating_cell_id: "c1", evaluation_queue: []},
+                  "s2" => %{evaluating_cell_id: nil, evaluation_queue: ["c2"]}
+                }
+              }, []} = Data.apply_operation(data, operation)
+    end
+
+    test "queues previous unevaluated and stale cells" do
+      data =
+        data_after_operations!([
+          {:insert_section, self(), 0, "s1"},
+          {:insert_cell, self(), "s1", 0, :elixir, "c1"},
+          {:insert_cell, self(), "s1", 1, :elixir, "c2"},
+          {:insert_section, self(), 1, "s2"},
+          {:insert_cell, self(), "s2", 0, :elixir, "c3"},
+          {:insert_cell, self(), "s2", 1, :elixir, "c4"},
+          # Evaluate first 2 cells
+          {:queue_cell_evaluation, self(), "c1"},
+          {:add_cell_evaluation_response, self(), "c1", {:ok, [1, 2, 3]}},
+          {:queue_cell_evaluation, self(), "c2"},
+          {:add_cell_evaluation_response, self(), "c2", {:ok, [1, 2, 3]}},
+          # Evaluate the first cell, so the second becomes stale
+          {:queue_cell_evaluation, self(), "c1"},
+          {:add_cell_evaluation_response, self(), "c1", {:ok, [1, 2, 3]}}
+        ])
+
+      # The above leads to:
+      #
+      # Section 1:
+      # * cell 1 - evaluated
+      # * cell 2 - stale
+      # Section 2:
+      # * cell 3 - fresh
+      # * cell 4 - fresh
+      #
+      # Queuing cell 4 should also queue cell 3 and cell 2,
+      # so that they all become evaluated.
+
+      operation = {:queue_cell_evaluation, self(), "c4"}
+
+      assert {:ok,
+              %{
+                cell_infos: %{
+                  "c2" => %{evaluation_status: :evaluating},
+                  "c3" => %{evaluation_status: :queued},
+                  "c4" => %{evaluation_status: :queued}
+                },
+                section_infos: %{
+                  "s1" => %{evaluating_cell_id: "c2", evaluation_queue: []},
+                  "s2" => %{evaluating_cell_id: nil, evaluation_queue: ["c3", "c4"]}
+                }
+              }, _actions} = Data.apply_operation(data, operation)
+    end
   end
 
   describe "apply_operation/2 given :add_cell_evaluation_stdout" do
@@ -567,6 +661,31 @@ defmodule Livebook.Session.DataTest do
                Data.apply_operation(data, operation)
     end
 
+    test "marks next queued cell in a further section as evaluating if there is one" do
+      data =
+        data_after_operations!([
+          {:insert_section, self(), 0, "s1"},
+          {:insert_cell, self(), "s1", 0, :elixir, "c1"},
+          {:insert_section, self(), 1, "s2"},
+          {:insert_cell, self(), "s2", 0, :elixir, "c2"},
+          {:queue_cell_evaluation, self(), "c1"},
+          {:queue_cell_evaluation, self(), "c2"}
+        ])
+
+      operation = {:add_cell_evaluation_response, self(), "c1", {:ok, [1, 2, 3]}}
+
+      assert {:ok,
+              %{
+                cell_infos: %{"c2" => %{evaluation_status: :evaluating}},
+                section_infos: %{
+                  "s1" => %{evaluating_cell_id: nil, evaluation_queue: []},
+                  "s2" => %{evaluating_cell_id: "c2", evaluation_queue: []}
+                }
+              },
+              [{:start_evaluation, %{id: "c2"}, %{id: "s2"}}]} =
+               Data.apply_operation(data, operation)
+    end
+
     test "if parent cells are not executed, marks them for evaluation first" do
       data =
         data_after_operations!([
@@ -595,11 +714,15 @@ defmodule Livebook.Session.DataTest do
           {:insert_section, self(), 0, "s1"},
           {:insert_cell, self(), "s1", 0, :elixir, "c1"},
           {:insert_cell, self(), "s1", 1, :elixir, "c2"},
-          # Evaluate both cells
+          {:insert_section, self(), 1, "s2"},
+          {:insert_cell, self(), "s2", 0, :elixir, "c3"},
+          # Evaluate all cells
           {:queue_cell_evaluation, self(), "c1"},
           {:add_cell_evaluation_response, self(), "c1", {:ok, [1, 2, 3]}},
           {:queue_cell_evaluation, self(), "c2"},
           {:add_cell_evaluation_response, self(), "c2", {:ok, [1, 2, 3]}},
+          {:queue_cell_evaluation, self(), "c3"},
+          {:add_cell_evaluation_response, self(), "c3", {:ok, [1, 2, 3]}},
           # Queue the first cell again
           {:queue_cell_evaluation, self(), "c1"}
         ])
@@ -608,7 +731,10 @@ defmodule Livebook.Session.DataTest do
 
       assert {:ok,
               %{
-                cell_infos: %{"c2" => %{validity_status: :stale}}
+                cell_infos: %{
+                  "c2" => %{validity_status: :stale},
+                  "c3" => %{validity_status: :stale}
+                }
               }, []} = Data.apply_operation(data, operation)
     end
   end
@@ -633,26 +759,32 @@ defmodule Livebook.Session.DataTest do
       assert :error = Data.apply_operation(data, operation)
     end
 
-    test "if the cell is evaluating, clears the corresponding section evaluation and the queue" do
+    test "if the cell is evaluating, clears all sections evaluation and queues" do
       data =
         data_after_operations!([
           {:insert_section, self(), 0, "s1"},
           {:insert_cell, self(), "s1", 0, :elixir, "c1"},
           {:insert_cell, self(), "s1", 1, :elixir, "c2"},
+          {:insert_section, self(), 1, "s2"},
+          {:insert_cell, self(), "s2", 0, :elixir, "c3"},
           {:queue_cell_evaluation, self(), "c1"},
-          {:queue_cell_evaluation, self(), "c2"}
+          {:add_cell_evaluation_response, self(), "c1", {:ok, [1, 2, 3]}},
+          {:queue_cell_evaluation, self(), "c2"},
+          {:queue_cell_evaluation, self(), "c3"}
         ])
 
-      operation = {:cancel_cell_evaluation, self(), "c1"}
+      operation = {:cancel_cell_evaluation, self(), "c2"}
 
       assert {:ok,
               %{
                 cell_infos: %{
-                  "c1" => %{validity_status: :fresh},
-                  "c2" => %{validity_status: :fresh}
+                  "c1" => %{validity_status: :fresh, evaluation_status: :ready},
+                  "c2" => %{validity_status: :fresh, evaluation_status: :ready},
+                  "c3" => %{validity_status: :fresh, evaluation_status: :ready}
                 },
                 section_infos: %{
-                  "s1" => %{evaluating_cell_id: nil, evaluation_queue: []}
+                  "s1" => %{evaluating_cell_id: nil, evaluation_queue: []},
+                  "s2" => %{evaluating_cell_id: nil, evaluation_queue: []}
                 }
               }, _actions} = Data.apply_operation(data, operation)
     end