diff --git a/Modules/selectmodule.c b/Modules/selectmodule.c
--- a/Modules/selectmodule.c
+++ b/Modules/selectmodule.c
@@ -1058,7 +1058,9 @@
 
 typedef struct {
     PyObject_HEAD
-    SOCKET epfd;                        /* epoll control file descriptor */
+    SOCKET epfd;                    /* epoll control file descriptor */
+    int maxevents;                  /* maximum number of epoll events */
+    struct epoll_event *evs;        /* epoll events buffer */
 } pyEpoll_Object;
 
 static PyTypeObject pyEpoll_Type;
@@ -1078,6 +1080,8 @@
     if (self->epfd >= 0) {
         int epfd = self->epfd;
         self->epfd = -1;
+        self->maxevents = 0;
+        self->evs = NULL;
         Py_BEGIN_ALLOW_THREADS
         if (close(epfd) < 0)
             save_errno = errno;
@@ -1114,6 +1118,15 @@
         PyErr_SetFromErrno(PyExc_OSError);
         return NULL;
     }
+
+    self->maxevents = FD_SETSIZE;
+    self->evs = PyMem_New(struct epoll_event, self->maxevents);
+    if (self->evs == NULL) {
+        Py_DECREF(self);
+        PyErr_NoMemory();
+        return NULL;
+    }
+ 
     return (PyObject *)self;
 }
 
@@ -1140,6 +1153,7 @@
 pyepoll_dealloc(pyEpoll_Object *self)
 {
     (void)pyepoll_internal_close(self);
+    PyMem_Free(self->evs);
     Py_TYPE(self)->tp_free(self);
 }
 
@@ -1317,10 +1331,9 @@
 {
     double dtimeout = -1.;
     int timeout;
-    int maxevents = -1;
+    int maxevents = self->maxevents;
     int nfds, i;
     PyObject *elist = NULL, *etuple = NULL;
-    struct epoll_event *evs = NULL;
     static char *kwlist[] = {"timeout", "maxevents", NULL};
 
     if (self->epfd < 0)
@@ -1343,25 +1356,25 @@
         timeout = (int)(dtimeout * 1000.0);
     }
 
-    if (maxevents == -1) {
-        maxevents = FD_SETSIZE-1;
-    }
-    else if (maxevents < 1) {
+    if (maxevents < 1) {
         PyErr_Format(PyExc_ValueError,
                      "maxevents must be greater than 0, got %d",
                      maxevents);
         return NULL;
-    }
+    } else if (maxevents != self->maxevents) {
+        struct epoll_event *orig_evs = self->evs;
 
-    evs = PyMem_New(struct epoll_event, maxevents);
-    if (evs == NULL) {
-        Py_DECREF(self);
-        PyErr_NoMemory();
-        return NULL;
+        PyMem_RESIZE(self->evs, struct epoll_event, maxevents);
+        if (self->evs == NULL) {
+            self->evs = orig_evs;
+            PyErr_NoMemory();
+            return NULL;
+        }
+        self->maxevents = maxevents;
     }
 
     Py_BEGIN_ALLOW_THREADS
-    nfds = epoll_wait(self->epfd, evs, maxevents, timeout);
+    nfds = epoll_wait(self->epfd, self->evs, self->maxevents, timeout);
     Py_END_ALLOW_THREADS
     if (nfds < 0) {
         PyErr_SetFromErrno(PyExc_OSError);
@@ -1374,7 +1387,7 @@
     }
 
     for (i = 0; i < nfds; i++) {
-        etuple = Py_BuildValue("iI", evs[i].data.fd, evs[i].events);
+        etuple = Py_BuildValue("iI", self->evs[i].data.fd, self->evs[i].events);
         if (etuple == NULL) {
             Py_CLEAR(elist);
             goto error;
@@ -1383,7 +1396,6 @@
     }
 
     error:
-    PyMem_Free(evs);
     return elist;
 }