diff --git a/synapse/federation/send_queue.py b/synapse/federation/send_queue.py
index 93e5acebc1..945832283f 100644
--- a/synapse/federation/send_queue.py
+++ b/synapse/federation/send_queue.py
@@ -35,7 +35,7 @@ from synapse.storage.presence import UserPresenceState
 from synapse.util.metrics import Measure
 import synapse.metrics
 
-from blist import sorteddict
+from sortedcontainers import SortedDict
 from collections import namedtuple
 
 import logging
@@ -56,19 +56,19 @@ class FederationRemoteSendQueue(object):
         self.is_mine_id = hs.is_mine_id
 
         self.presence_map = {}  # Pending presence map user_id -> UserPresenceState
-        self.presence_changed = sorteddict()  # Stream position -> user_id
+        self.presence_changed = SortedDict()  # Stream position -> user_id
 
         self.keyed_edu = {}  # (destination, key) -> EDU
-        self.keyed_edu_changed = sorteddict()  # stream position -> (destination, key)
+        self.keyed_edu_changed = SortedDict()  # stream position -> (destination, key)
 
-        self.edus = sorteddict()  # stream position -> Edu
+        self.edus = SortedDict()  # stream position -> Edu
 
-        self.failures = sorteddict()  # stream position -> (destination, Failure)
+        self.failures = SortedDict()  # stream position -> (destination, Failure)
 
-        self.device_messages = sorteddict()  # stream position -> destination
+        self.device_messages = SortedDict()  # stream position -> destination
 
         self.pos = 1
-        self.pos_time = sorteddict()
+        self.pos_time = SortedDict()
 
         # EVERYTHING IS SAD. In particular, python only makes new scopes when
         # we make a new function, so we need to make a new function so the inner
diff --git a/synapse/metrics/__init__.py b/synapse/metrics/__init__.py
index 50d99d7a5c..2ed82b04a5 100644
--- a/synapse/metrics/__init__.py
+++ b/synapse/metrics/__init__.py
@@ -17,6 +17,7 @@ import logging
 import functools
 import time
 import gc
+import platform
 
 from twisted.internet import reactor
 
@@ -30,6 +31,7 @@ from .process_collector import register_process_collector
 logger = logging.getLogger(__name__)
 
 
+running_on_pypy = platform.python_implementation() == 'PyPy'
 all_metrics = []
 all_collectors = []
 
@@ -174,6 +176,9 @@ def runUntilCurrentTimer(func):
         tick_time.inc_by(end - start)
         pending_calls_metric.inc_by(num_pending)
 
+        if running_on_pypy:
+            return ret
+
         # Check if we need to do a manual GC (since its been disabled), and do
         # one if necessary.
         threshold = gc.get_threshold()
@@ -206,6 +211,7 @@ try:
 
     # We manually run the GC each reactor tick so that we can get some metrics
     # about time spent doing GC,
-    gc.disable()
+    if not running_on_pypy:
+        gc.disable()
 except AttributeError:
     pass
diff --git a/synapse/python_dependencies.py b/synapse/python_dependencies.py
index 40eedb63cb..f9596bddaf 100644
--- a/synapse/python_dependencies.py
+++ b/synapse/python_dependencies.py
@@ -34,8 +34,8 @@ REQUIREMENTS = {
     "bcrypt": ["bcrypt>=3.1.0"],
     "pillow": ["PIL"],
     "pydenticon": ["pydenticon"],
-    "blist": ["blist"],
     "pysaml2>=3.0.0": ["saml2>=3.0.0"],
+    "sortedcontainers": ["sortedcontainers"],
     "pymacaroons-pynacl": ["pymacaroons"],
     "msgpack-python>=0.3.0": ["msgpack"],
     "phonenumbers>=8.2.0": ["phonenumbers"],
diff --git a/synapse/storage/_base.py b/synapse/storage/_base.py
index 2fbebd4907..2262776ab2 100644
--- a/synapse/storage/_base.py
+++ b/synapse/storage/_base.py
@@ -376,7 +376,7 @@ class SQLBaseStore(object):
         Returns:
             A list of dicts where the key is the column header.
         """
-        col_headers = list(intern(column[0]) for column in cursor.description)
+        col_headers = list(intern(str(column[0])) for column in cursor.description)
         results = list(
             dict(zip(col_headers, row)) for row in cursor
         )
diff --git a/synapse/storage/engines/__init__.py b/synapse/storage/engines/__init__.py
index 338b495611..8c868ece75 100644
--- a/synapse/storage/engines/__init__.py
+++ b/synapse/storage/engines/__init__.py
@@ -18,6 +18,7 @@ from .postgres import PostgresEngine
 from .sqlite3 import Sqlite3Engine
 
 import importlib
+import platform
 
 
 SUPPORTED_MODULE = {
@@ -31,6 +32,10 @@ def create_engine(database_config):
     engine_class = SUPPORTED_MODULE.get(name, None)
 
     if engine_class:
+        # pypy requires psycopg2cffi rather than psycopg2
+        if (name == "psycopg2" and
+                platform.python_implementation() == "PyPy"):
+            name = "psycopg2cffi"
         module = importlib.import_module(name)
         return engine_class(module, database_config)
 
diff --git a/synapse/types.py b/synapse/types.py
index 7cb24cecb2..cc7c182a78 100644
--- a/synapse/types.py
+++ b/synapse/types.py
@@ -169,7 +169,7 @@ class DomainSpecificString(
         except Exception:
             return False
 
-    __str__ = to_string
+    __repr__ = to_string
 
 
 class UserID(DomainSpecificString):
diff --git a/synapse/util/caches/stream_change_cache.py b/synapse/util/caches/stream_change_cache.py
index 941d873ab8..2ff46090a6 100644
--- a/synapse/util/caches/stream_change_cache.py
+++ b/synapse/util/caches/stream_change_cache.py
@@ -16,7 +16,7 @@
 from synapse.util.caches import register_cache, CACHE_SIZE_FACTOR
 
 
-from blist import sorteddict
+from sortedcontainers import SortedDict
 import logging
 
 
@@ -35,7 +35,7 @@ class StreamChangeCache(object):
     def __init__(self, name, current_stream_pos, max_size=10000, prefilled_cache={}):
         self._max_size = int(max_size * CACHE_SIZE_FACTOR)
         self._entity_to_key = {}
-        self._cache = sorteddict()
+        self._cache = SortedDict()
         self._earliest_known_stream_pos = current_stream_pos
         self.name = name
         self.metrics = register_cache(self.name, self._cache)