diff --git a/asyncpg/connect_utils.py b/asyncpg/connect_utils.py
index c65f68a6..5dd486d9 100644
--- a/asyncpg/connect_utils.py
+++ b/asyncpg/connect_utils.py
@@ -8,7 +8,7 @@
 
 import asyncio
 import collections
-from collections.abc import Callable
+from collections.abc import Callable, Sequence
 import enum
 import functools
 import getpass
@@ -41,10 +41,10 @@ class SSLMode(enum.IntEnum):
     verify_full = 5
 
     @classmethod
-    def parse(cls, sslmode):
-        if isinstance(sslmode, cls):
-            return sslmode
-        return getattr(cls, sslmode.replace('-', '_'))
+    def parse(cls, sslmode: typing.Union[str, SSLMode]) -> SSLMode:
+        if isinstance(sslmode, str):
+            return getattr(cls, sslmode.replace('-', '_'))
+        return sslmode
 
 
 class SSLNegotiation(compat.StrEnum):
@@ -52,20 +52,17 @@ class SSLNegotiation(compat.StrEnum):
     direct = "direct"
 
 
-_ConnectionParameters = collections.namedtuple(
-    'ConnectionParameters',
-    [
-        'user',
-        'password',
-        'database',
-        'ssl',
-        'sslmode',
-        'ssl_negotiation',
-        'server_settings',
-        'target_session_attrs',
-        'krbsrvname',
-        'gsslib',
-    ])
+class _ConnectionParameters(typing.NamedTuple):
+    user: str
+    password: typing.Optional[str]
+    database: str
+    ssl: typing.Union[ssl_module.SSLContext, bool, str, SSLMode, None]
+    sslmode: SSLMode
+    ssl_negotiation: SSLNegotiation
+    server_settings: typing.Optional[typing.Dict[str, str]]
+    target_session_attrs: SessionAttribute
+    krbsrvname: typing.Optional[str]
+    gsslib: str
 
 
 _ClientConfiguration = collections.namedtuple(
@@ -131,11 +128,13 @@ def _read_password_file(passfile: pathlib.Path) \
 
 
 def _read_password_from_pgpass(
-        *, passfile: typing.Optional[pathlib.Path],
-        hosts: typing.List[str],
-        ports: typing.List[int],
-        database: str,
-        user: str):
+    *,
+    passfile: pathlib.Path,
+    hosts: Sequence[str],
+    ports: typing.List[int],
+    database: str,
+    user: str
+) -> typing.Optional[str]:
     """Parse the pgpass file and return the matching password.
 
     :return:
@@ -167,7 +166,9 @@ def _read_password_from_pgpass(
     return None
 
 
-def _validate_port_spec(hosts, port):
+def _validate_port_spec(
+    hosts: Sequence[object], port: typing.Union[int, typing.List[int]]
+) -> typing.List[int]:
     if isinstance(port, list):
         # If there is a list of ports, its length must
         # match that of the host list.
@@ -181,15 +182,20 @@ def _validate_port_spec(hosts, port):
     return port
 
 
-def _parse_hostlist(hostlist, port, *, unquote=False):
+def _parse_hostlist(
+    hostlist: str,
+    port: typing.Union[int, typing.List[int]],
+    *,
+    unquote: bool = False,
+) -> typing.Tuple[typing.List[str], typing.List[int]]:
     if ',' in hostlist:
         # A comma-separated list of host addresses.
         hostspecs = hostlist.split(',')
     else:
         hostspecs = [hostlist]
 
-    hosts = []
-    hostlist_ports = []
+    hosts: typing.List[str] = []
+    hostlist_ports: typing.List[int] = []
 
     if not port:
         portspec = os.environ.get('PGPORT')
@@ -267,10 +273,25 @@ def _dot_postgresql_path(filename) -> typing.Optional[pathlib.Path]:
     return (homedir / '.postgresql' / filename).resolve()
 
 
-def _parse_connect_dsn_and_args(*, dsn, host, port, user,
-                                password, passfile, database, ssl,
-                                direct_tls, server_settings,
-                                target_session_attrs, krbsrvname, gsslib):
+def _parse_connect_dsn_and_args(
+    *,
+    dsn: str,
+    host: typing.Union[str, typing.List[str], typing.Tuple[str]],
+    port: typing.Union[int, typing.List[int]],
+    user: typing.Optional[str],
+    password: typing.Optional[str],
+    passfile: typing.Union[str, pathlib.Path, None],
+    database: typing.Optional[str],
+    ssl: typing.Union[bool, None, str, SSLMode],
+    direct_tls: typing.Optional[bool],
+    server_settings: typing.Optional[typing.Dict[str, str]],
+    target_session_attrs: typing.Optional[str],
+    krbsrvname: typing.Optional[str],
+    gsslib: typing.Optional[str],
+) -> typing.Tuple[
+    typing.List[typing.Union[str, typing.Tuple[str, int]]],
+    _ConnectionParameters,
+]:
     # `auth_hosts` is the version of host information for the purposes
     # of reading the pgpass file.
     auth_hosts = None
@@ -316,10 +337,12 @@ def _parse_connect_dsn_and_args(*, dsn, host, port, user,
             password = urllib.parse.unquote(dsn_password)
 
         if parsed.query:
-            query = urllib.parse.parse_qs(parsed.query, strict_parsing=True)
-            for key, val in query.items():
-                if isinstance(val, list):
-                    query[key] = val[-1]
+            query = {
+                key: val[-1]
+                for key, val in urllib.parse.parse_qs(
+                    parsed.query, strict_parsing=True
+                ).items()
+            }
 
             if 'port' in query:
                 val = query.pop('port')
@@ -491,7 +514,7 @@ def _parse_connect_dsn_and_args(*, dsn, host, port, user,
                 database=database, user=user,
                 passfile=passfile)
 
-    addrs = []
+    addrs: typing.List[typing.Union[str, typing.Tuple[str, int]]] = []
     have_tcp_addrs = False
     for h, p in zip(host, port):
         if h.startswith('/'):