diff --git a/tests/test_basic.py b/tests/test_basic.py
index 5f2dd4c..03b923d 100644
--- a/tests/test_basic.py
+++ b/tests/test_basic.py
@@ -241,6 +241,11 @@ def test_type_generic(self):
         assert issubclass(typing.Type[typing.List[int]], typing.Type[typing.Sequence[int]])
         assert not issubclass(typing.Type[typing.List[int]], typing.Type[typing.Sequence[str]])
 
+        assert issubclass(typing.Type[int], typing.Type[object])
+        assert issubclass(typing.Type[int], typing.Type[typing.Any])
+        assert not issubclass(typing.Type[object], typing.Type[int])
+        assert issubclass(typing.Type[typing.Any], typing.Type[int])
+
     def test_any(self):
         assert is_subtype(int, Any)
         assert is_subtype(Any, int)
@@ -863,6 +868,15 @@ class A:
         self.assertRaises(TypeError, A)
         self.assertRaises(TypeError, A, 2)
 
+    def test_self_reference(self):
+        @dataclass
+        class A:
+            items: List["A"]
+
+        a1 = A([])
+        a2 = A([a1])
+        self.assertRaises(TypeError, A, [1])
+
     def test_forward_references(self):
         @dataclass
         class A:
diff --git a/tests/test_pytypes.py b/tests/test_pytypes.py
new file mode 100644
index 0000000..371004b
--- /dev/null
+++ b/tests/test_pytypes.py
@@ -0,0 +1,320 @@
+import sys
+import unittest
+from unittest import TestCase
+import typing
+import collections.abc as cabc
+import io
+
+from runtype.base_types import DataType, GenericType, PhantomType, Variance
+from runtype.pytypes import type_caster, List, Dict, Int, Any, All, Constraint, String, Tuple, Iter, Literal, NoneType, Sequence, Mapping    
+from runtype.typesystem import TypeSystem
+
+make_type = type_caster.to_canon
+
+class TestTypes(TestCase):
+    def test_basic_types(self):
+        Int = DataType()
+        Str = DataType()
+        Array = GenericType(DataType(), Any, Variance.Covariant)
+
+        assert Int == Int
+        assert Int != Str
+        assert Int != Array
+        assert Int <= Any
+        assert Array <= Any
+
+        assert Array[Any] == Array
+        assert Array[Int] <= Array[Any]
+
+        array = Array[Array]
+        assert array[Array[Array]] <= array
+
+        self.assertRaises(TypeError, lambda: Int[Str])
+        self.assertRaises(TypeError, lambda: (Array[Array])[Int])
+
+        assert Int <= Int + Array
+        assert Int * Array == Int * Array
+
+    def test_phantom(self):
+        Int = DataType()
+        P = PhantomType()
+        Q = PhantomType()
+
+        assert P == P
+        assert P <= P
+        assert P != Q
+        assert not P <= Q
+        assert not Q <= P
+
+
+        assert Int <= P[Int]
+        assert P[Int] <= Int
+        assert P[Int] <= P[Int]
+        assert P[Int] <= P
+        assert not P <= P[Int]
+
+        assert P[Q] <= Q
+        assert P[Q] <= P
+        assert not P <= Q[Int]
+
+        assert P[Q[Int]] <= P
+        assert P[Q[Int]] <= Q
+        assert P[Q[Int]] <= Int
+        assert P[Q[Int]] <= P[Q]
+        assert P[Q[Int]] <= P[Int]
+        assert P[Q[Int]] <= Q[Int]
+        assert P[Q[Int]] <= P[Q[Int]]
+        assert Int <= P[Q[Int]]
+
+        assert P <= P + Int
+        assert not P <= Dict
+        assert not P <= Int + Dict
+
+
+    def test_pytypes1(self):
+        assert List + Dict == Dict + List
+        assert Any + ((Any + Any) + Any) is Any
+
+        assert (List+Dict) + Int == List + (Dict+Int)
+        assert (List+Dict) != 1
+        assert List + List == List
+
+        self.assertRaises(TypeError, lambda: 1 <= List)
+        self.assertRaises(TypeError, lambda: 1 >= List)
+        self.assertRaises(TypeError, lambda: 1 >= Any)
+        self.assertRaises(TypeError, lambda: 1 <= Any)
+        self.assertRaises(TypeError, lambda: 1 <= List+Dict)
+        self.assertRaises(TypeError, lambda: 1 >= List+Dict)
+        self.assertRaises(TypeError, lambda: 1 <= List*Dict)
+        self.assertRaises(TypeError, lambda: 1 >= List*Dict)
+
+        assert List[int] == List[int]
+        assert List[int] != List[str]
+        assert Dict == Dict[Any*Any]
+
+        assert repr(List[int]) == repr(List[int])
+        assert repr(Any) == 'Any'
+
+        assert List <= List + Dict
+        assert List + Dict >= List
+
+        assert {List+Dict: True}[Dict+List]		# test hashing
+
+        assert Dict*List <= Dict*List
+
+        # assert ((Int * Dict) * List) == (Int * (Dict * List))
+
+        assert List[Any] == List
+
+    def test_constraint(self):
+        int_pair = Constraint(typing.Sequence[int], [lambda a: len(a) == 2])
+        assert int_pair.test_instance([1,2])
+        assert not int_pair.test_instance([1,2,3])
+        assert not int_pair.test_instance([1,'a'])
+
+        assert String.test_instance('a')
+        assert not String.test_instance(3)
+
+        assert String(max_length=5).test_instance('abc')
+        assert String(min_length=2).test_instance('abc')
+        assert not String(max_length=5).test_instance('abcdef')
+        assert not String(min_length=5).test_instance('abc')
+
+        i = Int(min=10, max=12)
+        assert i.test_instance(11)
+        assert not i.test_instance(9)
+        assert not i.test_instance(13)
+
+        assert int_pair == int_pair
+        assert int_pair <= int_pair
+        assert int_pair >= int_pair
+        assert int_pair <= Any
+        assert Any >= int_pair
+        assert not int_pair <= Dict
+        assert not int_pair <= Int
+        assert not int_pair <= Int + Dict
+        assert not int_pair <= Tuple
+
+        assert int_pair <= Sequence
+        assert Sequence >= int_pair
+        assert int_pair <= Sequence[Int]
+        assert Sequence[Int] >= int_pair
+        assert not int_pair <= Sequence[String]
+        assert not Sequence[String] >= int_pair
+
+
+
+    def test_typesystem(self):
+        t = TypeSystem()
+        o = object()
+        assert t.to_canonical_type(o) is o
+
+        class IntOrder(TypeSystem):
+            def issubclass(self, a, b):
+                return a <= b 
+
+            def get_type(self, a):
+                return a
+
+        i = IntOrder()
+        assert i.isinstance(3, 3)
+        assert i.isinstance(3, 4)
+        assert not i.isinstance(4, 3)
+
+    def test_pytypes2(self):
+        assert Tuple <= Tuple
+        assert Tuple >= Tuple
+        # assert Tuple[int] <= Tuple
+        assert not List <= Tuple
+        assert not Tuple <= List
+        assert not Tuple <= Int
+        assert not Int <= Tuple
+
+        one = Literal([1])
+        one_two = Literal([1, 2])
+        one_three = Literal([1, 3])
+        assert one <= one_two
+        assert not one_three <= one_two
+        assert one_three >= one
+        assert not Literal([1]) <= Tuple
+        assert not Literal([1]) >= Tuple
+        assert not Tuple <= Literal([1])
+
+        Tuple.validate_instance((1, 2))
+        self.assertRaises(TypeError, Tuple.validate_instance, 1)
+
+        assert List[int] == List[int]
+
+        self.assertRaises(TypeError, lambda: Tuple >= 1)
+        self.assertRaises(TypeError, lambda: Literal >= 1)
+
+        assert type_caster.to_canon(typing.List[int]).cast_from([]) == []
+        assert type_caster.to_canon(typing.List[int]).cast_from(()) == []
+        assert type_caster.to_canon(typing.Dict[int, int]).cast_from({}) == {}
+        assert type_caster.to_canon(typing.Dict[int, int]).cast_from([]) == {}
+
+        tpl0 = type_caster.to_canon(typing.Tuple)
+        tpl1 = type_caster.to_canon(typing.Tuple[int])
+        tpl2 = type_caster.to_canon(typing.Tuple[int, ...])
+        tpl0b = type_caster.to_canon(tuple)
+        tpl3 = type_caster.to_canon(typing.Tuple[typing.Union[int, str]])
+        tpl4 = type_caster.to_canon(typing.Tuple[typing.Union[int, str], ...])
+        assert tpl0 is tpl0b
+        assert tpl1 <= tpl0
+        assert tpl2 <= tpl0
+
+        assert tpl3 <= tpl0
+        assert tpl1 <= tpl3
+        assert not tpl3 <= tpl1
+        assert not tpl0 <= tpl3
+
+        assert tpl2 <= tpl4
+
+        assert tpl2.test_instance((1,2,3))
+        assert not tpl2.test_instance((1,2,3, 'a'))
+        if sys.version_info >= (3, 11):
+            self.assertRaises(ValueError, type_caster.to_canon, typing.Tuple[...])
+            self.assertRaises(ValueError, type_caster.to_canon, typing.Tuple[int, str, ...])
+
+
+    def test_pytypes3(self):
+        assert Any + Int != Any
+        assert Int + Any != Any
+        assert Int + Any == Any + Int
+
+        assert All + Int == All
+        assert Int + All == All
+
+
+    def test_canonize_pytypes(self):
+        pytypes = [
+            int, str, list, dict, typing.Optional[int],
+            typing.Sequence[int],
+
+            # collections.abc
+            cabc.Hashable, cabc.Sized, cabc.Callable, cabc.Iterable, cabc.Container,
+            cabc.Collection, cabc.Iterator, cabc.Reversible, cabc.Generator,
+            cabc.Sequence, cabc.MutableSequence, cabc.ByteString,
+            cabc.Set, cabc.MutableSet,
+            cabc.Mapping, cabc.MutableMapping,
+            cabc.MappingView, cabc.ItemsView, cabc.KeysView, cabc.ValuesView,
+            cabc.Awaitable, cabc.Coroutine, 
+            cabc.AsyncIterable, cabc.AsyncIterator, cabc.AsyncGenerator,
+
+            typing.NoReturn
+        ]
+        for t in pytypes:
+            a = type_caster.to_canon(t)
+            # assert a.kernel == t, (a,t)
+
+
+        type_to_values = {
+            cabc.Hashable: ([1, "a", frozenset()], [{}, set()]),
+            cabc.Sized: ([(), {}], [10]),
+            cabc.Callable: ([int, lambda:1], [3, "a"]),
+            cabc.Iterable: ([(), {}, "", iter([])], [3]),
+            cabc.Container: ([(), ""], [3]),
+            cabc.Collection: ([(), ""], [iter([])]),
+            cabc.Iterator: ([iter([])], [[]]),
+            cabc.Reversible: ([[], ""], [iter([])]),
+            cabc.Generator: ([(x for x in [])], [3, iter('')]),
+
+            cabc.Set: ([set()], [{}]),
+            cabc.ItemsView: ([{}.items()], [{}])
+        }
+
+        for pyt, (good, bad) in type_to_values.items():
+            t = type_caster.to_canon(pyt)
+            for g in good:
+                assert t.test_instance(g), (t, g)
+            for b in bad:
+                assert not t.test_instance(b), (t, b)
+
+    def test_any(self):
+        assert Any <= Any
+        assert Any <= Any + Int
+        assert Any <= Any + NoneType
+        assert Any + Int <= Any
+        assert Any + NoneType <= Any
+
+    def test_invariance(self):
+        assert List <= Sequence
+        assert not List[List] <= List[Sequence]
+        assert not List[Sequence] <= List[List] 
+
+        assert Dict <= Mapping
+        assert Dict[Int, Int] <= Mapping[Int, Int]
+        assert Mapping[Int, List] <= Mapping[Int, Sequence]
+        assert not Dict[Int, List] <= Dict[Int, Sequence]
+        assert not Mapping[Int, Sequence] <= Mapping[Int, List]
+        assert not Dict[Int, Sequence] <= Dict[Int, List]
+
+    def test_callable(self):
+        repeat = make_type(typing.Callable[[str, int], str])
+        class _Str(str):
+            pass
+        assert repeat <= repeat
+        assert not make_type(typing.Callable[[_Str, int], str]) <= repeat
+        assert not make_type(typing.Callable[[int, str], str]) <= repeat
+        assert not repeat <= make_type(typing.Callable[[str, int], _Str])
+        assert make_type(typing.Callable[[str, int], _Str]) <= repeat
+        assert repeat <= make_type(typing.Callable[[_Str, int], str])
+
+    def test_io(self):
+        IO = make_type(io.IOBase)
+        TextIO = make_type(io.TextIOBase)
+        assert IO <= IO
+        assert TextIO <= IO
+        assert IO.test_instance(sys.stdout)
+
+    def test_typing_io(self):
+        IO = make_type(typing.IO)
+        TextIO = make_type(typing.TextIO)
+        assert IO <= IO
+        assert TextIO <= IO
+        assert IO.test_instance(sys.stdout)
+
+
+
+if __name__ == '__main__':
+    unittest.main()
\ No newline at end of file