diff --git a/changelog/new_add_new_rails_migration_class_name_cop.md b/changelog/new_add_new_rails_migration_class_name_cop.md
new file mode 100644
index 0000000000..09e3591de2
--- /dev/null
+++ b/changelog/new_add_new_rails_migration_class_name_cop.md
@@ -0,0 +1 @@
+* [#1044](https://github.com/rubocop/rubocop-rails/pull/1044): Add new `Rails/MigrationClassName` cop. ([@sambostock][])
diff --git a/config/default.yml b/config/default.yml
index 0fef0c30d5..70131373ef 100644
--- a/config/default.yml
+++ b/config/default.yml
@@ -656,6 +656,13 @@ Rails/MigrationClassName:
   Include:
     - db/**/*.rb
 
+Rails/MigrationTimestamp:
+  Description: 'Checks that migration filenames start with a valid timestamp.'
+  Enabled: pending
+  VersionAdded: '<<next>>'
+  Include:
+    - db/migrate/**/*.rb
+
 Rails/NegateInclude:
   Description: 'Prefer `collection.exclude?(obj)` over `!collection.include?(obj)`.'
   StyleGuide: 'https://rails.rubystyle.guide#exclude'
diff --git a/lib/rubocop/cop/rails/migration_timestamp.rb b/lib/rubocop/cop/rails/migration_timestamp.rb
new file mode 100644
index 0000000000..1ed8109461
--- /dev/null
+++ b/lib/rubocop/cop/rails/migration_timestamp.rb
@@ -0,0 +1,56 @@
+# frozen_string_literal: true
+
+require 'time'
+
+module RuboCop
+  module Cop
+    module Rails
+      # Checks that migration file names start with a valid timestamp.
+      #
+      # @example
+      #   # bad
+      #   # db/migrate/bad.rb
+
+      #   # bad
+      #   # db/migrate/123_bad.rb
+
+      #   # bad
+      #   # db/migrate/20171301000000_bad.rb
+      #
+      #   # good
+      #   # db/migrate/20170101000000_good.rb
+      #
+      class MigrationTimestamp < Base
+        include RangeHelp
+
+        MSG = 'Migration file name must start with `YYYYmmddHHMMSS_` timestamp.'
+
+        def on_new_investigation
+          file_path = processed_source.file_path
+          return unless file_path.include?('db/migrate')
+
+          timestamp = File.basename(file_path)[/\A\d{14}(?=_)/]
+          return if valid_timestamp?(timestamp)
+
+          add_offense(source_range(processed_source.buffer, 1, 0))
+        end
+
+        private
+
+        def valid_timestamp?(timestamp, format: '%Y%m%d%H%M%S')
+          timestamp &&
+            (time = Time.strptime(timestamp, format)) &&
+            # Time.strptime fuzzily accepts invalid dates around boundaries
+            #          | Wrong Days per Month | 24th Hour      | 60th Minute    | 60th Second
+            # ---------+----------------------+----------------+----------------+----------------
+            # Actual   |       20000231000000 | 20000101240000 | 20000101006000 | 20000101000060
+            # Expected |       20000302000000 | 20000102000000 | 20000101010000 | 20000101000100
+            # We want normalized values, so we can check if Time#strftime matches the original.
+            time.strftime(format) == timestamp
+        rescue ArgumentError
+          false
+        end
+      end
+    end
+  end
+end
diff --git a/lib/rubocop/cop/rails_cops.rb b/lib/rubocop/cop/rails_cops.rb
index 5e0db983d6..1a6e35a087 100644
--- a/lib/rubocop/cop/rails_cops.rb
+++ b/lib/rubocop/cop/rails_cops.rb
@@ -73,6 +73,7 @@
 require_relative 'rails/mailer_name'
 require_relative 'rails/match_route'
 require_relative 'rails/migration_class_name'
+require_relative 'rails/migration_timestamp'
 require_relative 'rails/negate_include'
 require_relative 'rails/not_null_column'
 require_relative 'rails/order_by_id'
diff --git a/spec/rubocop/cop/rails/migration_timestamp_spec.rb b/spec/rubocop/cop/rails/migration_timestamp_spec.rb
new file mode 100644
index 0000000000..ec2eae744e
--- /dev/null
+++ b/spec/rubocop/cop/rails/migration_timestamp_spec.rb
@@ -0,0 +1,65 @@
+# frozen_string_literal: true
+
+RSpec.describe RuboCop::Cop::Rails::MigrationTimestamp, :config do
+  it 'registers no offenses if timestamp is valid' do
+    expect_no_offenses(<<~RUBY, 'db/migrate/20170101000000_good.rb')
+      # ...
+    RUBY
+  end
+
+  it 'registers an offense if timestamp is impossible' do
+    expect_offense(<<~RUBY, 'db/migrate/20002222222222_bad.rb')
+      # ...
+      ^ Migration file name must start with `YYYYmmddHHMMSS_` timestamp.
+    RUBY
+  end
+
+  it 'registers an offense if timestamp swaps month and day' do
+    expect_offense(<<~RUBY, 'db/migrate/20003112000000_bad.rb')
+      # ...
+      ^ Migration file name must start with `YYYYmmddHHMMSS_` timestamp.
+    RUBY
+  end
+
+  it 'registers an offense if timestamp day is wrong' do
+    expect_offense(<<~RUBY, 'db/migrate/20000231000000_bad.rb')
+      # ...
+      ^ Migration file name must start with `YYYYmmddHHMMSS_` timestamp.
+    RUBY
+  end
+
+  it 'registers an offense if timestamp hours are invalid' do
+    expect_offense(<<~RUBY, 'db/migrate/20000101240000_bad.rb')
+      # ...
+      ^ Migration file name must start with `YYYYmmddHHMMSS_` timestamp.
+    RUBY
+  end
+
+  it 'registers an offense if timestamp minutes are invalid' do
+    expect_offense(<<~RUBY, 'db/migrate/20000101006000_bad.rb')
+      # ...
+      ^ Migration file name must start with `YYYYmmddHHMMSS_` timestamp.
+    RUBY
+  end
+
+  it 'registers an offense if timestamp seconds are invalid' do
+    expect_offense(<<~RUBY, 'db/migrate/20000101000060_bad.rb')
+      # ...
+      ^ Migration file name must start with `YYYYmmddHHMMSS_` timestamp.
+    RUBY
+  end
+
+  it 'registers an offense if timestamp is invalid' do
+    expect_offense(<<~RUBY, 'db/migrate/123_bad.rb')
+      # ...
+      ^ Migration file name must start with `YYYYmmddHHMMSS_` timestamp.
+    RUBY
+  end
+
+  it 'registers an offense if no timestamp at all' do
+    expect_offense(<<~RUBY, 'db/migrate/bad.rb')
+      # ...
+      ^ Migration file name must start with `YYYYmmddHHMMSS_` timestamp.
+    RUBY
+  end
+end