diff --git a/mypy/checkexpr.py b/mypy/checkexpr.py
index e850744b5c71..5092e5860947 100644
--- a/mypy/checkexpr.py
+++ b/mypy/checkexpr.py
@@ -2932,6 +2932,9 @@ def visit_index_with_type(self, left_type: Type, e: IndexExpr,
         # Visit the index, just to make sure we have a type for it available
         self.accept(index)
 
+        # Base method name for instance-based item access, might be changed for types:
+        method_name = '__getitem__'
+
         if isinstance(left_type, UnionType):
             original_type = original_type or left_type
             # Don't combine literal types, since we may need them for type narrowing.
@@ -2967,12 +2970,22 @@ def visit_index_with_type(self, left_type: Type, e: IndexExpr,
         elif (isinstance(left_type, TypeVarType)
               and not self.has_member(left_type.upper_bound, "__getitem__")):
             return self.visit_index_with_type(left_type.upper_bound, e, original_type)
-        else:
-            result, method_type = self.check_method_call_by_name(
-                '__getitem__', left_type, [e.index], [ARG_POS], e,
-                original_type=original_type)
-            e.method_type = method_type
-            return result
+        elif (isinstance(left_type, TypeType)
+                or (isinstance(left_type, CallableType) and left_type.is_type_obj())):
+            # When we do `SomeClass[1]`, we actually call `__class_getitem__`,
+            # not just `__getitem__`.
+            method_name = '__class_getitem__'
+
+        result, method_type = self.check_method_call_by_name(
+            method=method_name,
+            base_type=left_type,
+            args=[e.index],
+            arg_kinds=[ARG_POS],
+            context=e,
+            original_type=original_type,
+        )
+        e.method_type = method_type
+        return result
 
     def visit_tuple_slice_helper(self, left_type: TupleType, slic: SliceExpr) -> Type:
         begin: Sequence[Optional[int]] = [None]
diff --git a/mypy/checkmember.py b/mypy/checkmember.py
index 3efc39753627..9e4edf2f5989 100644
--- a/mypy/checkmember.py
+++ b/mypy/checkmember.py
@@ -235,7 +235,7 @@ def analyze_type_callable_member_access(name: str,
     if isinstance(ret_type, TupleType):
         ret_type = tuple_fallback(ret_type)
     if isinstance(ret_type, Instance):
-        if not mx.is_operator:
+        if not mx.is_operator or name == '__class_getitem__':
             # When Python sees an operator (eg `3 == 4`), it automatically translates that
             # into something like `int.__eq__(3, 4)` instead of `(3).__eq__(4)` as an
             # optimization.
@@ -250,6 +250,9 @@ def analyze_type_callable_member_access(name: str,
             # the corresponding method in the current instance to avoid this edge case.
             # See https://github.com/python/mypy/pull/1787 for more info.
             # TODO: do not rely on same type variables being present in all constructor overloads.
+
+            # We also allow `SomeClass[1]` acccess via `__class_getitem__`,
+            # it is very special. It only works this way for non-generic types.
             result = analyze_class_attribute_access(ret_type, name, mx,
                                                     original_vars=typ.items[0].variables)
             if result:
diff --git a/mypy/typeanal.py b/mypy/typeanal.py
index d400c7e1ca69..6dc9f6ae0b61 100644
--- a/mypy/typeanal.py
+++ b/mypy/typeanal.py
@@ -1072,6 +1072,12 @@ def fix_instance(t: Instance, fail: MsgCallback, note: MsgCallback,
 
     Also emit a suitable error if this is not due to implicit Any's.
     """
+    if not t.type.is_generic() and t.type.has_readable_member('__class_getitem__'):
+        # Corner case: we might have a non-generic class with `__class_getitem__`
+        # which is used for something else: not type application.
+        # So, in this case: we allow using this type without type arguments.
+        return
+
     if len(t.args) == 0:
         if use_generic_error:
             fullname: Optional[str] = None
@@ -1081,6 +1087,7 @@ def fix_instance(t: Instance, fail: MsgCallback, note: MsgCallback,
                                    unexpanded_type)
         t.args = (any_type,) * len(t.type.type_vars)
         return
+
     # Invalid number of type parameters.
     n = len(t.type.type_vars)
     s = '{} type arguments'.format(n)