diff --git a/core/kernel.rbs b/core/kernel.rbs
index b50e58111..0eca04bea 100644
--- a/core/kernel.rbs
+++ b/core/kernel.rbs
@@ -179,9 +179,9 @@ module Kernel : BasicObject
   #     c(4)   #=> []
   #     c(5)   #=> nil
   #
-  def self?.caller: (Integer start_or_range, ?Integer length) -> ::Array[String]?
-                  | (::Range[Integer] start_or_range) -> ::Array[String]?
-                  | () -> ::Array[String]
+  def self?.caller: () -> Array[String]
+                  | (int start, ?int? length) -> Array[String]?
+                  | (range[int] range) -> Array[String]?
 
   # <!--
   #   rdoc-file=vm_backtrace.c
@@ -204,8 +204,9 @@ module Kernel : BasicObject
   # Optionally you can pass a range, which will return an array containing the
   # entries within the specified range.
   #
-  def self?.caller_locations: (?Integer start_or_range, ?Integer length) -> ::Array[Thread::Backtrace::Location]?
-                            | (?::Range[Integer] start_or_range) -> ::Array[Thread::Backtrace::Location]?
+  def self?.caller_locations: () -> Array[Thread::Backtrace::Location]
+                            | (int start, ?int? length) -> Array[Thread::Backtrace::Location]?
+                            | (range[int] range) -> Array[Thread::Backtrace::Location]?
 
   # <!--
   #   rdoc-file=vm_eval.c
@@ -310,6 +311,16 @@ module Kernel : BasicObject
   #
   def self?.block_given?: () -> bool
 
+  alias self.iterator? self.block_given?
+
+  # <!--
+  #   rdoc-file=vm_eval.c
+  #   - iterator?      -> true or false
+  # -->
+  # Deprecated.  Use block_given? instead.
+  #
+  alias iterator? block_given?
+
   # <!--
   #   rdoc-file=vm_eval.c
   #   - local_variables    -> array
@@ -322,7 +333,7 @@ module Kernel : BasicObject
   #     end
   #     local_variables   #=> [:fred, :i]
   #
-  def self?.local_variables: () -> ::Array[Symbol]
+  def self?.local_variables: () -> Array[Symbol]
 
   # <!--
   #   rdoc-file=random.c
@@ -777,7 +788,7 @@ module Kernel : BasicObject
   # If *const* is defined as autoload, the file name to be loaded is replaced with
   # *filename*.  If *const* is defined but not as autoload, does nothing.
   #
-  def self?.autoload: (interned _module, String filename) -> NilClass
+  def self?.autoload: (interned const, path filename) -> nil
 
   # <!--
   #   rdoc-file=load.c
@@ -801,7 +812,7 @@ module Kernel : BasicObject
   #       autoload?(:B)          #=> "b"
   #     end
   #
-  def self?.autoload?: (interned name) -> String?
+  def self?.autoload?: (interned name, ?boolish inherit) -> String?
 
   # <!--
   #   rdoc-file=proc.c
@@ -1175,7 +1186,7 @@ module Kernel : BasicObject
   #
   #     global_variables.grep /std/   #=> [:$stdin, :$stdout, :$stderr]
   #
-  def self?.global_variables: () -> ::Array[Symbol]
+  def self?.global_variables: () -> Array[Symbol]
 
   # <!--
   #   rdoc-file=load.c
@@ -1826,7 +1837,12 @@ module Kernel : BasicObject
   # ------------|---------------------------------------------
   # <tt>'-'</tt>|Whether the entities exist and are identical.
   #
-  def self?.test: (String | Integer cmd, String | IO file1, ?String | IO file2) -> (TrueClass | FalseClass | Time | nil | Integer)
+  def self?.test: ('b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'G' | 'k' | 'l' | 'o' | 'O' | 'p' | 'r' | 'R' | 'S' | 'u' | 'w' | 'W' | 'x' | 'X' | 'z' |
+                    98 | 99 | 100 | 101 | 102 | 103 | 71 | 107 | 108 | 111 | 79 | 112 | 114 | 82 | 83 | 117 | 119 | 87 | 120 | 88 | 122, path filepath) -> bool
+                | ('s' | 115, path filepath) -> Integer?
+                | ('A' | 'M' | 'C' | 65 | 77 | 67, path filepath) -> Time
+                | ('<' | '=' | '>' | '-' | 60 | 61 | 62 | 45, path filepath1, path filepath2) -> bool
+                | (String | int cmd, path filepath1, ?path filepath2) -> (bool | Time | Integer | nil)
 
   # <!--
   #   rdoc-file=vm_eval.c
diff --git a/lib/rbs/test/type_check.rb b/lib/rbs/test/type_check.rb
index 980d5dd2d..40e6288ee 100644
--- a/lib/rbs/test/type_check.rb
+++ b/lib/rbs/test/type_check.rb
@@ -340,7 +340,12 @@ def value(val, type)
         when Types::Variable
           true
         when Types::Literal
-          type.literal == val
+          begin
+            type.literal == val
+          rescue NoMethodError
+            raise if defined?(val.==)
+            false
+          end
         when Types::Union
           type.types.any? {|type| value(val, type) }
         when Types::Intersection
diff --git a/test/stdlib/Kernel_test.rb b/test/stdlib/Kernel_test.rb
index 1af2bf0dd..94d786ea2 100644
--- a/test/stdlib/Kernel_test.rb
+++ b/test/stdlib/Kernel_test.rb
@@ -81,17 +81,239 @@ def test_String
                      Kernel, :String, ToS.new
   end
 
+  TOPLEVEL___callee__ = __callee__ # outside of a method
+  def test___callee__
+    assert_send_type '() -> Symbol',
+                     Kernel, :__callee__
+    assert_type 'nil', TOPLEVEL___callee__
+  end
+
+  TOPLEVEL___method__ = __method__ # outside of a method
+  def test___method__
+    assert_send_type '() -> Symbol',
+                     Kernel, :__method__
+    assert_type 'nil', TOPLEVEL___method__
+  end
+
+  def test___dir__
+    assert_send_type '() -> String',
+                     Kernel, :__dir__
+
+    # Make sure it can return `nil`; this can't go through `assert_send_type`,
+    # as it's only `nil` thru `eval`s
+    assert_equal nil, eval('__dir__')
+  end
+
+  def test_autoload
+    with_interned :TestModuleForAutoload do |const|
+      with_path '/does/not/exist' do |path|
+        assert_send_type '(interned, path) -> nil',
+                         Kernel, :autoload, const, path
+      end
+    end
+  end
+
   def test_autoload?
-    with_interned :TestModuleForAutoload do |interned|
-      assert_send_type "(::interned) -> String?",
-                       Kernel, :autoload?, interned
+    with_interned :TestModuleForAutoloadP do |const|
+      assert_send_type '(interned) -> nil',
+                       Kernel, :autoload?, const
+
+      with_boolish do |inherit|
+        assert_send_type '(interned, boolish) -> nil',
+                         Kernel, :autoload?, const, inherit
+      end
+    end
+
+    # Unfortunately, `autoload` doesn't play well with `assert_send_type`
+    Kernel.autoload :TestModuleForAutoloadP, '/does/not/exist'
+
+    with_interned :TestModuleForAutoloadP do |const|
+      assert_type 'String', Kernel.autoload?(const)
+
+      with_boolish do |inherit|
+        assert_type 'String', Kernel.autoload?(const, inherit)
+      end
+    end
+  end
+
+  def test_binding
+    assert_send_type '() -> Binding',
+                     Kernel, :binding
+  end
+
+  def test_block_given?(method: :block_given?)
+    assert_send_type '() -> bool',
+                     Kernel, method
+  end
+
+  def test_iterator?
+    silence_warning :deprecated do
+      test_block_given?(method: :iterator?)
+    end
+  end
+
+  def test_caller
+    assert_send_type '() -> Array[String]',
+                     Kernel, :caller
+
+    with_int 1 do |start|
+      assert_send_type '(int) -> Array[String]',
+                       Kernel, :caller, start
+
+      with_int(2).and_nil do |length|
+        assert_send_type '(int, int?) -> Array[String]',
+                         Kernel, :caller, start, length
+      end
+    end
+
+    with_int 100000 do |start|
+      assert_send_type '(int) -> nil',
+                       Kernel, :caller, start
+
+      with_int(2).and_nil do |length|
+        assert_send_type '(int, int?) -> nil',
+                         Kernel, :caller, start, length
+      end
+    end
+
+    with_range with_int(1), with_int(2) do |range|
+      assert_send_type '(range[int]) -> Array[String]',
+                       Kernel, :caller, range
+    end
+
+    with_range with_int(100000) ,with_int(100001) do |range|
+      assert_send_type '(range[int]) -> nil',
+                       Kernel, :caller, range
     end
+  end
+
+  def test_caller_locations
+    assert_send_type '() -> Array[Thread::Backtrace::Location]',
+                     Kernel, :caller_locations
+
+    with_int 1 do |start|
+      assert_send_type '(int) -> Array[Thread::Backtrace::Location]',
+                       Kernel, :caller_locations, start
+
+      with_int(2).and_nil do |length|
+        assert_send_type '(int, int?) -> Array[Thread::Backtrace::Location]',
+                         Kernel, :caller_locations, start, length
+      end
+    end
+
+    with_int 100000 do |start|
+      assert_send_type '(int) -> nil',
+                       Kernel, :caller_locations, start
+
+      with_int(2).and_nil do |length|
+        assert_send_type '(int, int?) -> nil',
+                         Kernel, :caller_locations, start, length
+      end
+    end
+
+    with_range with_int(1), with_int(2) do |range|
+      assert_send_type '(range[int]) -> Array[Thread::Backtrace::Location]',
+                       Kernel, :caller_locations, range
+    end
+
+    with_range with_int(100000) ,with_int(100001) do |range|
+      assert_send_type '(range[int]) -> nil',
+                       Kernel, :caller_locations, range
+    end
+  end
+
+  def test_global_variables
+    assert_send_type '() -> Array[Symbol]',
+                     Kernel, :global_variables
+  end
+
+  def test_local_variables
+    assert_send_type '() -> Array[Symbol]',
+                     Kernel, :local_variables
+  end
+
+  def test_test
+    # true/false tests
+    with_path do |filepath|
+      %w[b c d e f g G k l o O p r R S u w W x X z].each do |test_char|
+        test_ord = test_char.ord
 
-    autoload :TestModuleForAutoload, '/shouldnt/be/executed'
+        with test_char, test_ord do |test_literal|
+          assert_send_type "('b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'G' | 'k' | 'l' | 'o' | 'O' | 'p' | 'r' | 'R' | 'S' | 'u' | 'w' | 'W' | 'x' | 'X' | 'z' |
+                              98 |  99 | 100 | 101 | 102 | 103 |  71 | 107 | 108 | 111 |  79 | 112 | 114 |  82 |  83 | 117 | 119 |  87 | 120 |  88 | 122, path) -> bool",
+                           Kernel, :test, test_literal, filepath
+        end
 
-    with_interned :TestModuleForAutoload do |interned|
-      assert_send_type "(::interned) -> String?",
-                       Kernel, :autoload?, interned
+        with_int(test_ord).and test_char do |test_nonliteral|
+          assert_send_type "(String | int, path, ?path) -> (bool | Time | Integer | nil)",
+                           Kernel, :test, test_nonliteral, filepath
+        end
+      end
+    end
+
+    # Integer? tests
+    %w[s].each do |test_char|
+      test_ord = test_char.ord
+
+      with_path __FILE__ do |filepath|
+        with test_char, test_ord do |test_literal|
+          assert_send_type "('s' | 115, path) -> Integer",
+                           Kernel, :test, test_literal, filepath
+        end
+
+        with_int(test_ord).and test_char do |test_nonliteral|
+          assert_send_type "(String | int, path, ?path) -> (bool | Time | Integer | nil)",
+                           Kernel, :test, test_nonliteral, filepath
+        end
+      end
+
+      with_path '/not/a/file' do |filepath|
+        with test_char, test_ord do |test_literal|
+          assert_send_type "('s' | 115, path) -> nil",
+                           Kernel, :test, test_literal, filepath
+        end
+
+        with_int(test_ord).and test_char do |test_nonliteral|
+          assert_send_type "(String | int, path, ?path) -> (bool | Time | Integer | nil)",
+                           Kernel, :test, test_nonliteral, filepath
+        end
+      end
+    end
+
+    # Time tests
+    with_path __FILE__ do |filepath|
+      %w[A M C].each do |test_char|
+        test_ord = test_char.ord
+
+        with test_char, test_ord do |test_literal|
+          assert_send_type "('A' | 'M' | 'C' | 65 | 77 | 67, path) -> Time",
+                           Kernel, :test, test_literal, filepath
+        end
+
+        with_int(test_ord).and test_char do |test_nonliteral|
+          assert_send_type "(String | int, path, ?path) -> (bool | Time | Integer | nil)",
+                           Kernel, :test, test_nonliteral, filepath
+        end
+      end
+    end
+
+    # Comparison Tests
+    with_path __dir__ + '/Integer_test.rb' do |filepath1|
+      with_path __FILE__ do |filepath2|
+        %w[< = > -].each do |test_char|
+          test_ord = test_char.ord
+
+          with test_char, test_ord do |test_literal|
+            assert_send_type "('<' | '=' | '>' | '-' | 60 | 61 | 62 | 45, path, path) -> bool",
+                             Kernel, :test, test_literal, filepath1, filepath2
+          end
+
+          with_int(test_ord).and test_char do |test_nonliteral|
+            assert_send_type "(String | int, path, ?path) -> (bool | Time | Integer | nil)",
+                             Kernel, :test, test_nonliteral, filepath1, filepath2
+          end
+        end
+      end
     end
   end
 
diff --git a/test/stdlib/test_helper.rb b/test/stdlib/test_helper.rb
index beea49803..204cbdecc 100644
--- a/test/stdlib/test_helper.rb
+++ b/test/stdlib/test_helper.rb
@@ -149,6 +149,14 @@ def self.included(base)
   end
 
   RUBY_EXECUTABLE = ENV["RUBY"] || RbConfig.ruby
+
+  def silence_warning(which)
+    old_warning = Warning[which]
+    Warning[which] = false
+    yield
+  ensure
+    Warning[which] = old_warning
+  end
 end
 
 class StdlibTest < Test::Unit::TestCase