diff --git a/marshmallow_dataclass/__init__.py b/marshmallow_dataclass/__init__.py
index aa47ab0..7ad8da5 100644
--- a/marshmallow_dataclass/__init__.py
+++ b/marshmallow_dataclass/__init__.py
@@ -482,17 +482,17 @@ def _field_by_supertype(
 
 def _generic_type_add_any(typ: type) -> type:
     """if typ is generic type without arguments, replace them by Any."""
-    if typ is list or typ is List:
+    if typ in (list, List):
         typ = List[Any]
-    elif typ is dict or typ is Dict:
+    elif typ in (dict, Dict):
         typ = Dict[Any, Any]
-    elif typ is Mapping:
+    elif typ in (Mapping, collections.abc.Mapping):
         typ = Mapping[Any, Any]
-    elif typ is Sequence:
+    elif typ in (Sequence, collections.abc.Sequence):
         typ = Sequence[Any]
-    elif typ is set or typ is Set:
+    elif typ in (set, Set, collections.abc.Set):
         typ = Set[Any]
-    elif typ is frozenset or typ is FrozenSet:
+    elif typ in (frozenset, FrozenSet):
         typ = FrozenSet[Any]
     return typ
 
diff --git a/tests/test_collection.py b/tests/test_collection.py
index 594e8cb..1b876e6 100644
--- a/tests/test_collection.py
+++ b/tests/test_collection.py
@@ -322,6 +322,39 @@ class AnyMapping:
         )
         self.assertEqual(schema.dump(loaded), data_in)
 
+    @unittest.skipIf(sys.version_info < (3, 9), "PEP 585 unsupported")
+    def test_collections_mapping_no_arg(self):
+        import collections.abc
+
+        @dataclass
+        class AnyMapping:
+            value: collections.abc.Mapping
+
+        schema = AnyMapping.Schema()
+
+        # can load a sequence of mixed kind
+        data_in = {
+            "value": {
+                1: "a number key a str value",
+                "a str key a number value": 2,
+                None: "this is still valid",
+                "even this": None,
+            }
+        }
+        loaded = schema.load(data_in)
+        self.assertEqual(
+            loaded,
+            AnyMapping(
+                value={
+                    1: "a number key a str value",
+                    "a str key a number value": 2,
+                    None: "this is still valid",
+                    "even this": None,
+                }
+            ),
+        )
+        self.assertEqual(schema.dump(loaded), data_in)
+
     def test_mapping_of_frozen_dataclass(self):
         @dataclass(frozen=True)
         class Elm: