diff --git a/ChangeLog b/ChangeLog
index f0cc1df31e..615941fb66 100644
--- a/ChangeLog
+++ b/ChangeLog
@@ -12,8 +12,13 @@ What's New in astroid 2.7.1?
 ============================
 Release date: TBA
 
+* When processing dataclass attributes, only do typing inference on collection types.
+  Support for instantiating other typing types is left for the future, if desired.
+
+
 * Fixed LookupMixIn missing from ``astroid.node_classes``.
 
+  Closes #1129
 
 
 What's New in astroid 2.7.0?
diff --git a/astroid/brain/brain_dataclasses.py b/astroid/brain/brain_dataclasses.py
index f42c981cad..b4be357fc9 100644
--- a/astroid/brain/brain_dataclasses.py
+++ b/astroid/brain/brain_dataclasses.py
@@ -97,16 +97,7 @@ def infer_dataclass_attribute(
     if value is not None:
         yield from value.infer(context=ctx)
     if annotation is not None:
-        klass = None
-        try:
-            klass = next(annotation.infer())
-        except (InferenceError, StopIteration):
-            yield Uninferable
-
-        if not isinstance(klass, ClassDef):
-            yield Uninferable
-        else:
-            yield klass.instantiate_class()
+        yield from _infer_instance_from_annotation(annotation, ctx=ctx)
     else:
         yield Uninferable
 
@@ -229,6 +220,44 @@ def _is_init_var(node: NodeNG) -> bool:
     return getattr(inferred, "name", "") == "InitVar"
 
 
+# Allowed typing classes for which we support inferring instances
+_INFERABLE_TYPING_TYPES = frozenset(
+    (
+        "Dict",
+        "FrozenSet",
+        "List",
+        "Set",
+        "Tuple",
+    )
+)
+
+
+def _infer_instance_from_annotation(
+    node: NodeNG, ctx: context.InferenceContext = None
+) -> Generator:
+    """Infer an instance corresponding to the type annotation represented by node.
+
+    Currently has limited support for the typing module.
+    """
+    klass = None
+    try:
+        klass = next(node.infer(context=ctx))
+    except (InferenceError, StopIteration):
+        yield Uninferable
+    if not isinstance(klass, ClassDef):
+        yield Uninferable
+    elif klass.root().name in (
+        "typing",
+        "",
+    ):  # "" because of synthetic nodes in brain_typing.py
+        if klass.name in _INFERABLE_TYPING_TYPES:
+            yield klass.instantiate_class()
+        else:
+            yield Uninferable
+    else:
+        yield klass.instantiate_class()
+
+
 if PY37_PLUS:
     AstroidManager().register_transform(
         ClassDef, dataclass_transform, is_decorated_with_dataclass
diff --git a/tests/unittest_brain_dataclasses.py b/tests/unittest_brain_dataclasses.py
index f7b08747f0..f90893d5cd 100644
--- a/tests/unittest_brain_dataclasses.py
+++ b/tests/unittest_brain_dataclasses.py
@@ -4,6 +4,7 @@
 from astroid import bases, nodes
 from astroid.const import PY37_PLUS
 from astroid.exceptions import InferenceError
+from astroid.util import Uninferable
 
 if not PY37_PLUS:
     pytest.skip("Dataclasses were added in 3.7", allow_module_level=True)
@@ -235,3 +236,62 @@ class A:
         assert len(inferred) == 1
         assert isinstance(inferred[0], nodes.Const)
         assert inferred[0].value == "hi"
+
+
+def test_inference_generic_collection_attribute():
+    """Test that an attribute with a generic collection type from the
+    typing module is inferred correctly.
+    """
+    attr_nodes = astroid.extract_node(
+        """
+    from dataclasses import dataclass, field
+    import typing
+
+    @dataclass
+    class A:
+        dict_prop: typing.Dict[str, str]
+        frozenset_prop: typing.FrozenSet[str]
+        list_prop: typing.List[str]
+        set_prop: typing.Set[str]
+        tuple_prop: typing.Tuple[int, str]
+
+    a = A({}, frozenset(), [], set(), (1, 'hi'))
+    a.dict_prop       #@
+    a.frozenset_prop  #@
+    a.list_prop       #@
+    a.set_prop        #@
+    a.tuple_prop      #@
+    """
+    )
+    names = (
+        "Dict",
+        "FrozenSet",
+        "List",
+        "Set",
+        "Tuple",
+    )
+    for node, name in zip(attr_nodes, names):
+        inferred = next(node.infer())
+        assert isinstance(inferred, bases.Instance)
+        assert inferred.name == name
+
+
+def test_inference_callable_attribute():
+    """Test that an attribute with a Callable annotation is inferred as Uninferable.
+
+    See issue#1129.
+    """
+    instance = astroid.extract_node(
+        """
+    from dataclasses import dataclass
+    from typing import Any, Callable
+
+    @dataclass
+    class A:
+        enabled: Callable[[Any], bool]
+
+    A(lambda x: x == 42).enabled  #@
+    """
+    )
+    inferred = next(instance.infer())
+    assert inferred is Uninferable