diff --git a/rest_framework/decorators.py b/rest_framework/decorators.py
index 3b572c09e..883a1c6d2 100644
--- a/rest_framework/decorators.py
+++ b/rest_framework/decorators.py
@@ -61,6 +61,9 @@ def api_view(http_method_names=None):
         WrappedAPIView.parser_classes = getattr(func, 'parser_classes',
                                                 APIView.parser_classes)
 
+        WrappedAPIView.middleware_classes = getattr(func, 'middleware_classes',
+                                                    APIView.middleware_classes)
+
         WrappedAPIView.authentication_classes = getattr(func, 'authentication_classes',
                                                         APIView.authentication_classes)
 
diff --git a/rest_framework/middleware.py b/rest_framework/middleware.py
new file mode 100644
index 000000000..06bfb815e
--- /dev/null
+++ b/rest_framework/middleware.py
@@ -0,0 +1,26 @@
+class BaseMiddleware:
+    """
+    All middleware classes should extend BaseMiddleware.
+    """
+
+    def process_request(self, request):
+        pass
+
+    def process_response(self, response):
+        pass
+
+
+class FooMiddleware(BaseMiddleware):
+    def process_request(self, request):
+        request._foo = "foo"
+
+    def process_response(self, response):
+        pass
+
+
+class BarMiddleware(BaseMiddleware):
+    def process_request(self, request):
+        pass
+
+    def process_response(self, response):
+        response._bar = "bar"
diff --git a/rest_framework/settings.py b/rest_framework/settings.py
index 96b664574..a60477f27 100644
--- a/rest_framework/settings.py
+++ b/rest_framework/settings.py
@@ -37,6 +37,10 @@ DEFAULTS = {
         'rest_framework.parsers.FormParser',
         'rest_framework.parsers.MultiPartParser'
     ],
+    'DEFAULT_MIDDLEWARE_CLASSES': [
+        'rest_framework.middleware.FooMiddleware',
+        'rest_framework.middleware.BarMiddleware'
+    ],
     'DEFAULT_AUTHENTICATION_CLASSES': [
         'rest_framework.authentication.SessionAuthentication',
         'rest_framework.authentication.BasicAuthentication'
@@ -133,6 +137,7 @@ DEFAULTS = {
 IMPORT_STRINGS = [
     'DEFAULT_RENDERER_CLASSES',
     'DEFAULT_PARSER_CLASSES',
+    'DEFAULT_MIDDLEWARE_CLASSES',
     'DEFAULT_AUTHENTICATION_CLASSES',
     'DEFAULT_PERMISSION_CLASSES',
     'DEFAULT_THROTTLE_CLASSES',
diff --git a/rest_framework/views.py b/rest_framework/views.py
index 4c30029fd..0903b2c76 100644
--- a/rest_framework/views.py
+++ b/rest_framework/views.py
@@ -106,6 +106,7 @@ class APIView(View):
     # The following policies may be set at either globally, or per-view.
     renderer_classes = api_settings.DEFAULT_RENDERER_CLASSES
     parser_classes = api_settings.DEFAULT_PARSER_CLASSES
+    middleware_classes = api_settings.DEFAULT_MIDDLEWARE_CLASSES
     authentication_classes = api_settings.DEFAULT_AUTHENTICATION_CLASSES
     throttle_classes = api_settings.DEFAULT_THROTTLE_CLASSES
     permission_classes = api_settings.DEFAULT_PERMISSION_CLASSES
@@ -265,6 +266,12 @@ class APIView(View):
         """
         return [parser() for parser in self.parser_classes]
 
+    def get_middlewares(self):
+        """
+        Instantiates and returns the list of middlewares that this view can use.
+        """
+        return [middleware() for middleware in self.middleware_classes]
+
     def get_authenticators(self):
         """
         Instantiates and returns the list of authenticators that this view can use.
@@ -479,6 +486,24 @@ class APIView(View):
             request.force_plaintext_errors(use_plaintext_traceback)
         raise exc
 
+    def process_request(self, request):
+        """
+        Pre-process the request by the middleware instances.
+        """
+        middlewares = getattr(self, "middlewares", self.get_middlewares())
+        for middleware in middlewares:
+            if hasattr(middleware, "process_request"):
+                middleware.process_request(request)
+
+    def process_response(self, response):
+        """
+        Pre-process the response by the middleware instances.
+        """
+        middlewares = getattr(self, "middlewares", self.get_middlewares())
+        for middleware in reversed(middlewares):
+            if hasattr(middleware, "process_response"):
+                middleware.process_response(response)
+
     # Note: Views are made CSRF exempt from within `as_view` as to prevent
     # accidental removal of this exemption in cases where `dispatch` needs to
     # be overridden.
@@ -492,8 +517,10 @@ class APIView(View):
         request = self.initialize_request(request, *args, **kwargs)
         self.request = request
         self.headers = self.default_response_headers  # deprecate?
+        self.middlewares = self.get_middlewares()
 
         try:
+            self.process_request(request)
             self.initial(request, *args, **kwargs)
 
             # Get the appropriate handler method
@@ -504,6 +531,7 @@ class APIView(View):
                 handler = self.http_method_not_allowed
 
             response = handler(request, *args, **kwargs)
+            self.process_response(response)
 
         except Exception as exc:
             response = self.handle_exception(exc)
diff --git a/tests/test_rest_middleware.py b/tests/test_rest_middleware.py
new file mode 100644
index 000000000..b8837ca75
--- /dev/null
+++ b/tests/test_rest_middleware.py
@@ -0,0 +1,71 @@
+from django.test import TestCase, override_settings
+from django.urls import path
+
+from rest_framework import status
+from rest_framework.middleware import (
+    BarMiddleware, BaseMiddleware, FooMiddleware
+)
+from rest_framework.response import Response
+from rest_framework.test import APIClient, APIRequestFactory
+from rest_framework.views import APIView
+
+factory = APIRequestFactory()
+
+
+class DummyRequestMiddleware(BaseMiddleware):
+    def process_request(self, request):
+        request._dummy = "dummy"
+
+
+class DummyResponseMiddleware(BarMiddleware):
+    def process_response(self, response):
+        response._dummy = "dummy"
+
+
+class MockView(APIView):
+    def get(self, request):
+        response = Response(status=status.HTTP_200_OK)
+        response._request = request  # test client sets `request` input
+        return response
+
+
+urlpatterns = [
+    path('foo/', MockView.as_view(middleware_classes=[FooMiddleware])),
+    path('bar/', MockView.as_view(middleware_classes=[BarMiddleware])),
+    path('multiple/', MockView.as_view(middleware_classes=[DummyRequestMiddleware, DummyResponseMiddleware])),
+    path('none/', MockView.as_view(middleware_classes=[]))
+]
+
+
+@override_settings(ROOT_URLCONF=__name__)
+class FooMiddlewareTests(TestCase):
+    def test_foo_middleware_process_request(self):
+        response = APIClient().get('/foo/')
+        request = response._request
+        assert getattr(request, "_foo") == "foo"
+        assert response.status_code == status.HTTP_200_OK
+
+
+@override_settings(ROOT_URLCONF=__name__)
+class BarMiddlewareTests(TestCase):
+    def test_bar_middleware_process_response(self):
+        response = APIClient().get('/bar/')
+        assert getattr(response, "_bar") == "bar"
+        assert response.status_code == status.HTTP_200_OK
+
+
+@override_settings(ROOT_URLCONF=__name__)
+class MultipleMiddlewareClassesTests(TestCase):
+    def test_multiple_middleware_classes_process_request_and_response(self):
+        response = APIClient().get('/multiple/')
+        request = response._request
+        assert getattr(request, "_dummy") == "dummy"
+        assert getattr(response, "_dummy") == "dummy"
+        assert response.status_code == status.HTTP_200_OK
+
+
+@override_settings(ROOT_URLCONF=__name__)
+class NoMiddlewareClassesTests(TestCase):
+    def test_bar_middleware_process_request(self):
+        response = APIClient().get('/none/')
+        assert response.status_code == status.HTTP_200_OK