diff --git a/docs/source/reference-hazmat.rst b/docs/source/reference-hazmat.rst
index 8fd0a85a94..1ba6725351 100644
--- a/docs/source/reference-hazmat.rst
+++ b/docs/source/reference-hazmat.rst
@@ -194,6 +194,30 @@ Unix-like systems provide the following functions:
        become writable.
 
 
+.. function:: wait_priority(fd)
+   :async:
+
+   Block until there is an "exceptional condition" on the given file
+   descriptor.
+
+   .. warning::
+
+      This method uses the ``POLLPRI`` flag to the ``epoll`` system call.
+      Its semantics are highly specific to the type of file descriptor
+      you're passing to it.
+
+   This test may be used to
+   * wait for changes to hardware inputs (Linux GPIO)
+   * notice changed state of a PTY slave by the master
+   * …
+
+   :arg fd:
+       integer file descriptor, or else an object with a ``fileno()`` method
+   :raises trio.ResourceBusyError:
+       if another task is already waiting for an exceptional condition on
+       the given fd.
+
+
 Kqueue-specific API
 -------------------
 
diff --git a/trio/_core/_io_epoll.py b/trio/_core/_io_epoll.py
index da605dd701..1d425c1c51 100644
--- a/trio/_core/_io_epoll.py
+++ b/trio/_core/_io_epoll.py
@@ -9,6 +9,7 @@
 class _EpollStatistics:
     tasks_waiting_read = attr.ib()
     tasks_waiting_write = attr.ib()
+    tasks_waiting_prio = attr.ib()
     backend = attr.ib(default="epoll")
 
 
@@ -16,6 +17,7 @@ class _EpollStatistics:
 class EpollWaiters:
     read_task = attr.ib(default=None)
     write_task = attr.ib(default=None)
+    prio_task = attr.ib(default=None)
 
     def flags(self):
         flags = 0
@@ -23,6 +25,8 @@ def flags(self):
             flags |= select.EPOLLIN
         if self.write_task is not None:
             flags |= select.EPOLLOUT
+        if self.prio_task is not None:
+            flags |= select.EPOLLPRI
         if not flags:
             return None
         # XX not sure if EPOLLEXCLUSIVE is actually safe... I think
@@ -47,14 +51,18 @@ class EpollIOManager:
     def statistics(self):
         tasks_waiting_read = 0
         tasks_waiting_write = 0
+        tasks_waiting_prio = 0
         for waiter in self._registered.values():
             if waiter.read_task is not None:
                 tasks_waiting_read += 1
             if waiter.write_task is not None:
                 tasks_waiting_write += 1
+            if waiter.prio_task is not None:
+                tasks_waiting_prio += 1
         return _EpollStatistics(
             tasks_waiting_read=tasks_waiting_read,
             tasks_waiting_write=tasks_waiting_write,
+            tasks_waiting_prio=tasks_waiting_prio,
         )
 
     def close(self):
@@ -70,12 +78,19 @@ def handle_io(self, timeout):
             # Clever hack stolen from selectors.EpollSelector: an event
             # with EPOLLHUP or EPOLLERR flags wakes both readers and
             # writers.
-            if flags & ~select.EPOLLIN and waiters.write_task is not None:
-                _core.reschedule(waiters.write_task)
-                waiters.write_task = None
-            if flags & ~select.EPOLLOUT and waiters.read_task is not None:
+            if flags & ~(
+                select.EPOLLOUT | select.EPOLLPRI
+            ) and waiters.read_task is not None:
                 _core.reschedule(waiters.read_task)
                 waiters.read_task = None
+            if flags & ~(
+                select.EPOLLIN | select.EPOLLPRI
+            ) and waiters.write_task is not None:
+                _core.reschedule(waiters.write_task)
+                waiters.write_task = None
+            if flags & select.EPOLLPRI and waiters.prio_task is not None:
+                _core.reschedule(waiters.prio_task)
+                waiters.prio_task = None
             self._update_registrations(fd, True)
 
     def _update_registrations(self, fd, currently_registered):
@@ -122,3 +137,7 @@ async def wait_readable(self, fd):
     @_public
     async def wait_writable(self, fd):
         await self._epoll_wait(fd, "write_task")
+
+    @_public
+    async def wait_priority(self, fd):
+        await self._epoll_wait(fd, "prio_task")
diff --git a/trio/hazmat.py b/trio/hazmat.py
index d57d6b3d3c..5c3f1b4ad5 100644
--- a/trio/hazmat.py
+++ b/trio/hazmat.py
@@ -22,6 +22,7 @@
     "add_instrument",
     "current_clock",
     "current_statistics",
+    "wait_priority",
     "wait_writable",
     "wait_readable",
     "ParkingLot",