diff --git a/dataclass_type_validator/__init__.py b/dataclass_type_validator/__init__.py
index ebf4e1a..5a09dca 100644
--- a/dataclass_type_validator/__init__.py
+++ b/dataclass_type_validator/__init__.py
@@ -5,6 +5,7 @@
 from typing import Any
 from typing import Optional
 from typing import Dict
+import types
 
 GlobalNS_T = Dict[str, Any]
 
@@ -109,15 +110,23 @@ def _validate_typing_literal(expected_type: type, value: Any, strict: bool) -> O
 
 _validate_typing_mappings = {
     'List': _validate_typing_list,
+    'list': _validate_typing_list,
     'Tuple': _validate_typing_tuple,
+    'tuple': _validate_typing_tuple,
     'FrozenSet': _validate_typing_frozenset,
+    'frozenset': _validate_typing_frozenset,
     'Dict': _validate_typing_dict,
+    'dict': _validate_typing_dict,
     'Callable': _validate_typing_callable,
 }
 
 
+def _type_name(t: type) -> str:
+    return t._name if hasattr(t, '_name') else t.__name__
+
+
 def _validate_sequential_types(expected_type: type, value: Any, strict: bool, globalns: GlobalNS_T) -> Optional[str]:
-    validate_func = _validate_typing_mappings.get(expected_type._name)
+    validate_func = _validate_typing_mappings.get(_type_name(expected_type))
     if validate_func is not None:
         return validate_func(expected_type, value, strict, globalns)
 
@@ -132,16 +141,22 @@ def _validate_sequential_types(expected_type: type, value: Any, strict: bool, gl
         return
 
     if strict:
-        raise RuntimeError(f'Unknown type of {expected_type} (_name = {expected_type._name})')
+        raise RuntimeError(f'Unknown type of {expected_type} (_name = {_type_name(expected_type)}')
 
+def _is_generic_alias(expected_type: type) -> bool:
+    if sys.version_info < (3, 9):
+        return isinstance(expected_type, typing._GenericAlias)
+    else:
+        return isinstance(expected_type, (typing._GenericAlias, types.GenericAlias))
 
-def _validate_types(expected_type: type, value: Any, strict: bool, globalns: GlobalNS_T) -> Optional[str]:
-    if isinstance(expected_type, type):
-        return _validate_type(expected_type=expected_type, value=value)
 
-    if isinstance(expected_type, typing._GenericAlias):
+def _validate_types(expected_type: type, value: Any, strict: bool, globalns: GlobalNS_T) -> Optional[str]:
+    if _is_generic_alias(expected_type):
         return _validate_sequential_types(expected_type=expected_type, value=value,
                                           strict=strict, globalns=globalns)
+    
+    if isinstance(expected_type, type):
+        return _validate_type(expected_type=expected_type, value=value)
 
     if isinstance(expected_type, typing.ForwardRef):
         referenced_type = _evaluate_forward_reference(expected_type, globalns)
diff --git a/tests/test_validator.py b/tests/test_validator.py
index b54d493..5614a3b 100644
--- a/tests/test_validator.py
+++ b/tests/test_validator.py
@@ -361,3 +361,77 @@ def optional_type_name(arg_type_name):
         return f"typing.Union\\[({arg_type_name}, NoneType|NoneType, {arg_type_name})\\]"
 
     return f"typing.Optional\\[{arg_type_name}\\]"
+
+# Tests for generic types, only in 3.9+
+if sys.version_info >= (3, 9):
+
+    @dataclasses.dataclass(frozen=True)
+    class DataclassTestGenericList:
+        array_of_numbers: list[int]
+        array_of_strings: list[str]
+        array_of_optional_strings: list[typing.Optional[str]]
+
+        def __post_init__(self):
+            dataclass_type_validator(self)
+
+
+    class TestTypeValidationGenericList:
+        def test_build_success(self):
+            assert isinstance(DataclassTestGenericList(
+                array_of_numbers=[],
+                array_of_strings=[],
+                array_of_optional_strings=[],
+            ), DataclassTestGenericList)
+            assert isinstance(DataclassTestGenericList(
+                array_of_numbers=[1, 2],
+                array_of_strings=['abc'],
+                array_of_optional_strings=['abc', None]
+            ), DataclassTestGenericList)
+
+        def test_build_failure_on_array_numbers(self):
+            with pytest.raises(TypeValidationError, match='must be an instance of list\\[int\\]'):
+                assert isinstance(DataclassTestGenericList(
+                    array_of_numbers=['abc'],
+                    array_of_strings=['abc'],
+                    array_of_optional_strings=['abc', None]
+                ), DataclassTestGenericList)
+
+        def test_build_failure_on_array_strings(self):
+            with pytest.raises(TypeValidationError, match='must be an instance of list\\[str\\]'):
+                assert isinstance(DataclassTestGenericList(
+                    array_of_numbers=[1, 2],
+                    array_of_strings=[123],
+                    array_of_optional_strings=['abc', None]
+                ), DataclassTestGenericList)
+
+        def test_build_failure_on_array_optional_strings(self):
+            with pytest.raises(TypeValidationError,
+                               match=f"must be an instance of list\\[{optional_type_name('str')}\\]"):
+                assert isinstance(DataclassTestGenericList(
+                    array_of_numbers=[1, 2],
+                    array_of_strings=['abc'],
+                    array_of_optional_strings=[123, None]
+                ), DataclassTestGenericList)
+
+    @dataclasses.dataclass(frozen=True)
+    class DataclassTestGenericDict:
+        str_to_str: dict[str, str]
+        str_to_any: dict[str, typing.Any]
+
+        def __post_init__(self):
+            dataclass_type_validator(self)
+
+
+    class TestTypeValidationGenericDict:
+        def test_build_success(self):
+            assert isinstance(DataclassTestGenericDict(
+                str_to_str={'str': 'str'},
+                str_to_any={'str': 'str', 'str2': 123}
+            ), DataclassTestGenericDict)
+
+        def test_build_failure(self):
+            with pytest.raises(TypeValidationError, match='must be an instance of dict\\[str, str\\]'):
+                assert isinstance(DataclassTestGenericDict(
+                    str_to_str={'str': 123},
+                    str_to_any={'key': []}
+                ), DataclassTestGenericDict)