diff --git a/examples/eio/dune b/examples/eio/dune
index 1b01b8b..cf34605 100644
--- a/examples/eio/dune
+++ b/examples/eio/dune
@@ -1,6 +1,11 @@
 (executables
  (libraries httpun httpun-eio httpun_examples base stdio eio_main eio-ssl)
- (names eio_echo_post eio_get eio_ssl_get)
+ (names
+  eio_echo_post
+  eio_get
+  eio_ssl_get
+  eio_connect_server
+  eio_connect_client)
  (flags :standard -warn-error -A))
 
 (alias
diff --git a/examples/eio/eio_connect_client.ml b/examples/eio/eio_connect_client.ml
new file mode 100644
index 0000000..d0e2f21
--- /dev/null
+++ b/examples/eio/eio_connect_client.ml
@@ -0,0 +1,102 @@
+module Arg = Stdlib.Arg
+open Httpun
+
+let http_handler ~on_eof response response_body =
+  match response with
+  | { Response.status = `OK; _ } as response ->
+    Format.eprintf "response: %a@." Response.pp_hum response;
+    let rec on_read bs ~off ~len =
+      Bigstringaf.substring ~off ~len bs |> print_string;
+      flush stdout;
+      Body.Reader.schedule_read response_body ~on_read ~on_eof
+    in
+    Body.Reader.schedule_read response_body ~on_read ~on_eof
+  | response ->
+    Format.fprintf Format.err_formatter "%a\n%!" Response.pp_hum response;
+    Stdlib.exit 124
+
+let proxy_handler _env ~sw ~headers flow ~on_eof response _response_body =
+  Format.eprintf "CONNECT response: %a@." Response.pp_hum response;
+  match response with
+  | { Response.status = `OK; _ } as response ->
+    (* This means we can now communicate via any protocol on the socket since
+       the server approved the tunnel.
+
+       We'll be boring and use HTTP/1.1 again. *)
+    let connection = Httpun_eio.Client.create_connection ~sw flow in
+    let exit_cond = Eio.Condition.create () in
+    Eio.Fiber.fork ~sw (fun () ->
+      let response_handler =
+        http_handler ~on_eof:(fun () ->
+          Stdlib.Format.eprintf "http eof@.";
+          Eio.Condition.broadcast exit_cond;
+          on_eof ())
+      in
+      let request_body =
+        Httpun_eio.Client.request
+          ~flush_headers_immediately:true
+          ~error_handler:Httpun_examples.Client.error_handler
+          ~response_handler
+          connection
+          (Request.create ~headers `GET "/")
+      in
+      Body.Writer.close request_body);
+    Eio.Condition.await_no_mutex exit_cond;
+    Httpun_eio.Client.shutdown connection |> Eio.Promise.await
+  | _response -> Stdlib.exit 124
+
+let main port proxy_host =
+  let real_host = "example.com:80" in
+  Eio_main.run (fun _env ->
+    Eio.Switch.run (fun sw ->
+      let fd = Unix.socket ~cloexec:true Unix.PF_INET Unix.SOCK_STREAM 0 in
+      let addrs =
+        Eio_unix.run_in_systhread (fun () ->
+          Unix.getaddrinfo
+            proxy_host
+            (Int.to_string port)
+            [ Unix.(AI_FAMILY PF_INET) ])
+      in
+      Eio_unix.run_in_systhread (fun () ->
+        Unix.connect fd (List.hd addrs).ai_addr);
+      let socket = Eio_unix.Net.import_socket_stream ~sw ~close_unix:true fd in
+      let headers = Headers.of_list [ "host", real_host ] in
+      let connection = Httpun_eio.Client.create_connection ~sw socket in
+
+      let exit_cond = Eio.Condition.create () in
+      Eio.Fiber.fork ~sw (fun ()->
+      let response_handler =
+        fun response response_body ->
+          Eio.Fiber.fork ~sw @@ fun () ->
+          proxy_handler _env ~sw socket ~headers ~on_eof:(fun () ->
+            Stdlib.Format.eprintf "(connect) eof@.";
+            Eio.Condition.broadcast exit_cond)
+          response
+          response_body
+      in
+      let request_body =
+        Httpun_eio.Client.request
+          ~flush_headers_immediately:true
+          ~error_handler:Httpun_examples.Client.error_handler
+          ~response_handler
+          connection
+          (Request.create ~headers `CONNECT real_host)
+      in
+      Body.Writer.close request_body;
+      Eio.Condition.await_no_mutex exit_cond;
+
+      Httpun_eio.Client.shutdown connection |> Eio.Promise.await)))
+
+let () =
+  let host = ref None in
+  let port = ref 80 in
+  Arg.parse
+    [ "-p", Set_int port, " Port number (80 by default)" ]
+    (fun host_argument -> host := Some host_argument)
+    "lwt_get.exe [-p N] HOST";
+  let host =
+    match !host with
+    | None -> failwith "No hostname provided"
+    | Some host -> host
+  in
+  main !port host
diff --git a/examples/eio/eio_connect_server.ml b/examples/eio/eio_connect_server.ml
new file mode 100644
index 0000000..eb8b141
--- /dev/null
+++ b/examples/eio/eio_connect_server.ml
@@ -0,0 +1,120 @@
+(* curl -v -p -x http://localhost:8080 http://example.com *)
+open Base
+module Arg = Stdlib.Arg
+open Httpun_eio
+open Httpun
+
+let error_handler (_ : Eio.Net.Sockaddr.stream) =
+  Httpun_examples.Server.error_handler
+
+let request_handler
+      env
+      ~sw
+      ~u
+      flow
+      (_ : Eio.Net.Sockaddr.stream)
+      { Gluten.reqd; _ }
+  =
+  match Reqd.request reqd with
+  | { Request.meth = `CONNECT; headers; _ } ->
+    Stdlib.Format.eprintf "x: %a@." Request.pp_hum (Reqd.request reqd);
+    let host, port =
+      let host_and_port = Headers.get_exn headers "host" in
+      let[@ocaml.warning "-8"] [ host; port ] =
+        String.split_on_chars ~on:[ ':' ] host_and_port
+      in
+      host, port
+    in
+    let () =
+      (* todo: try/with *)
+      let p, u' = Eio.Promise.create () in
+      Eio.Fiber.fork ~sw (fun () ->
+        Eio.Net.with_tcp_connect
+          (Eio.Stdenv.net env)
+          ~host
+          ~service:port
+          (fun upstream ->
+             Eio.Promise.resolve u' ();
+             Stdlib.Format.eprintf
+               "connected to upstream %s (port %s)@."
+               host
+               port;
+             Eio.Fiber.both
+               (fun () -> Eio.Flow.copy flow upstream)
+               (fun () -> Eio.Flow.copy upstream flow);
+             Eio.Promise.resolve_ok u ()));
+      Eio.Promise.await p
+    in
+    Reqd.respond_with_string reqd (Response.create `OK) ""
+  | _ ->
+    let headers = Headers.of_list [ "connection", "close" ] in
+    Reqd.respond_with_string
+      reqd
+      (Response.create ~headers `Method_not_allowed)
+      ""
+
+let log_connection_error ex =
+  Eio.traceln "Uncaught exception handling client: %a" Fmt.exn ex
+
+let main port =
+  Eio_main.run (fun env ->
+    let listen_address = `Tcp (Eio.Net.Ipaddr.V4.loopback, port) in
+    let network = Eio.Stdenv.net env in
+    let handler ~u =
+     fun ~sw client_addr socket ->
+      let request_handler = request_handler env ~sw ~u socket in
+      Server.create_connection_handler
+        ~request_handler
+        ~error_handler
+        ~sw
+        client_addr
+        socket
+    in
+    Eio.Switch.run (fun sw ->
+      let socket =
+        Eio.Net.listen
+          ~reuse_addr:true
+          ~reuse_port:false
+          ~backlog:5
+          ~sw
+          network
+          listen_address
+      in
+      Stdio.printf "Listening on port %i and echoing POST requests.\n" port;
+      Stdio.printf "To send a POST request, try one of the following\n\n";
+      Stdio.printf
+        "  echo \"Testing echo POST\" | dune exec examples/async/async_post.exe\n";
+      Stdio.printf
+        "  echo \"Testing echo POST\" | dune exec examples/lwt/lwt_post.exe\n";
+      Stdio.printf
+        "  echo \"Testing echo POST\" | curl -XPOST --data @- \
+         http://localhost:%d\n\n\
+         %!"
+        port;
+      let domain_mgr = Eio.Stdenv.domain_mgr env in
+      let p, _ = Eio.Promise.create () in
+      for _i = 1 to Stdlib.Domain.recommended_domain_count () do
+        Eio.Fiber.fork_daemon ~sw (fun () ->
+          Eio.Domain_manager.run domain_mgr (fun () ->
+            Eio.Switch.run (fun sw ->
+              while true do
+                Eio.Net.accept_fork
+                  socket
+                  ~sw
+                  ~on_error:log_connection_error
+                  (fun client_sock client_addr ->
+                     let p, u = Eio.Promise.create () in
+                     handler ~sw ~u client_addr client_sock;
+                     Eio.Promise.await_exn p)
+              done;
+              `Stop_daemon)))
+      done;
+      Eio.Promise.await p))
+
+let () =
+  let port = ref 8080 in
+  Arg.parse
+    [ "-p", Arg.Set_int port, " Listening port number (8080 by default)" ]
+    ignore
+    "Echoes POST requests. Runs forever.";
+  main !port
diff --git a/lib/reqd.ml b/lib/reqd.ml
index ef3c54e..b1929b5 100644
--- a/lib/reqd.ml
+++ b/lib/reqd.ml
@@ -257,8 +257,8 @@ let persistent_connection t =
   t.persistent
 
 let input_state t : Io_state.t =
-  match t.response_state with
-  | Upgrade _ -> Ready
+  match t.response_state, t.request.meth with
+  | Upgrade _,_ | _, `CONNECT -> Wait
   | _ ->
     if Body.Reader.is_closed t.request_body
     then Complete
@@ -266,8 +266,11 @@ let input_state t : Io_state.t =
     then Ready
     else Wait
 
-let output_state { response_state; writer; _ } =
-  Response_state.output_state response_state ~writer
+let output_state { request; response_state; writer; _ } =
+  Response_state.output_state
+    response_state
+    ~request_method:request.meth
+    ~writer
 
 let flush_request_body t =
   if Body.Reader.has_pending_output t.request_body
diff --git a/lib/respd.ml b/lib/respd.ml
index 7c0499e..ebc39ee 100644
--- a/lib/respd.ml
+++ b/lib/respd.ml
@@ -25,13 +25,18 @@ type t =
   ; mutable persistent : bool
   }
 
-let create error_handler request request_body writer response_handler =
+let create error_handler (request: Request.t) request_body writer response_handler =
   let rec handler response body =
     let t = Lazy.force t in
     if t.persistent then
       t.persistent <- Response.persistent_connection response;
-    let next_state : Request_state.t = match response.status with
-      | `Switching_protocols ->
+    let next_state : Request_state.t = match request.meth, response.status with
+      (* From RFC9110§6.4.1:
+       *   2xx (Successful) responses to a CONNECT request method (Section
+       *   9.3.6) switch the connection to tunnel mode instead of having
+       *   content. *)
+      | `CONNECT, #Status.successful
+      | _, `Switching_protocols ->
         Upgraded response
       | _ ->
         Received_response (response, body)
@@ -103,9 +108,7 @@ let input_state t : Io_state.t =
     else if Body.Reader.is_read_scheduled response_body
     then Ready
     else Wait
-    (* Upgraded is "Complete" because the descriptor doesn't wish to receive
-     * any more input. *)
-  | Upgraded _
+  | Upgraded _ -> Wait
   | Closed -> Complete
 
 let output_state { request_body; state; writer; _ } : Io_state.t =
diff --git a/lib/response_state.ml b/lib/response_state.ml
index a88f305..80c3ba7 100644
--- a/lib/response_state.ml
+++ b/lib/response_state.ml
@@ -4,18 +4,23 @@ type t =
   | Streaming of Response.t * Body.Writer.t
   | Upgrade of Response.t * (unit -> unit)
 
-let output_state t ~writer : Io_state.t =
-  match t with
-  | Fixed _ -> Complete
-  | Waiting ->
-    if Serialize.Writer.is_closed writer then Complete
-    else Wait
-  | Streaming(_, response_body) ->
-    if Serialize.Writer.is_closed writer then Complete
-    else if Body.Writer.requires_output response_body
-    then Ready
-    else Complete
-  | Upgrade _ -> Ready
+let output_state =
+  let response_sent_state = function
+    | `CONNECT -> Io_state.Wait
+    | _ -> Complete
+  in
+  fun t ~request_method ~writer : Io_state.t ->
+     match t with
+    | Upgrade _ -> Wait
+    | Waiting ->
+      if Serialize.Writer.is_closed writer then Complete
+      else Wait
+    | Fixed _ -> response_sent_state request_method
+    | Streaming(_, response_body) ->
+      if Serialize.Writer.is_closed writer then response_sent_state request_method
+      else if Body.Writer.requires_output response_body
+      then Ready
+      else response_sent_state request_method
 
 let flush_response_body t =
   match t with
diff --git a/lib/server_connection.ml b/lib/server_connection.ml
index e737627..3398517 100644
--- a/lib/server_connection.ml
+++ b/lib/server_connection.ml
@@ -338,8 +338,18 @@ let rec _next_write_operation t =
       if Reader.is_closed t.reader
       then shutdown t;
       Writer.next t.writer
-    | Error { response_state; _ } ->
-      match Response_state.output_state response_state ~writer:t.writer with
+    | Error { request; response_state } ->
+      match
+        let request_method =
+          Option.value
+            ~default:`GET
+            (Option.map (fun (t: Request.t) -> t.meth) request)
+        in
+        Response_state.output_state
+          response_state
+          ~request_method
+          ~writer:t.writer
+      with
       | Wait -> `Yield
       | Ready ->
         flush_response_error_body response_state;
diff --git a/lib_test/test_client_connection.ml b/lib_test/test_client_connection.ml
index 17e278c..f3e0a18 100644
--- a/lib_test/test_client_connection.ml
+++ b/lib_test/test_client_connection.ml
@@ -1866,6 +1866,41 @@ let test_read_response_before_shutdown () =
   connection_is_shutdown t;
 ;;
 
+let test_client_connect () =
+  let writer_woken_up = ref false in
+  let reader_woken_up = ref false in
+  let request' = Request.create
+    ~headers:(Headers.of_list ["host", "example.com:80"])
+    `CONNECT "/"
+  in
+  let t = create () in
+  let response = Response.create `OK in
+  let body =
+    request
+      t
+      request'
+      ~flush_headers_immediately:true
+      ~response_handler:(default_response_handler response)
+      ~error_handler:no_error_handler
+  in
+  write_request t request';
+  writer_yielded t;
+  Body.Writer.close body;
+  reader_ready t;
+  read_response t response;
+  reader_yielded t;
+  yield_reader t (fun () -> reader_woken_up := true);
+  writer_yielded t;
+  yield_writer t (fun () -> writer_woken_up := true);
+  Alcotest.(check bool) "Reader hasn't woken up yet" false !reader_woken_up;
+  Alcotest.(check bool) "Writer hasn't woken up yet" false !writer_woken_up;
+  shutdown t;
+  Alcotest.(check bool) "Reader woken up" true !reader_woken_up;
+  Alcotest.(check bool) "Writer woken up" true !writer_woken_up;
+  connection_is_shutdown t;
+;;
+
+
 let tests =
   [ "commit parse after every header line", `Quick, test_commit_parse_after_every_header
   ; "GET"         , `Quick, test_get
@@ -1914,4 +1949,5 @@ let tests =
   ; "shut down closes request body ", `Quick, test_read_response_before_shutdown
   ; "report exn during body read", `Quick, test_report_exn_during_body_read
   ; "read response after write eof", `Quick, test_can_read_response_after_write_eof
+  ; "Client support for CONNECT", `Quick, test_client_connect
   ]
diff --git a/lib_test/test_server_connection.ml b/lib_test/test_server_connection.ml
index c5a2aac..22a0863 100644
--- a/lib_test/test_server_connection.ml
+++ b/lib_test/test_server_connection.ml
@@ -957,7 +957,7 @@ let test_respond_with_upgrade () =
   read_request t (Request.create `GET "/");
   write_response ~msg:"Upgrade response written" t (Response.create `Switching_protocols);
   Alcotest.(check bool) "Callback was called" true !upgraded;
-  reader_ready t;
+  reader_yielded t;
 ;;
 
 let test_unexpected_eof () =
@@ -2438,6 +2438,24 @@ let test_write_response_after_read_eof () =
   connection_is_shutdown t;
 ;;
 
+let test_connect_method () =
+  let upgraded = ref false in
+  let upgrade_handler reqd =
+    Reqd.respond_with_upgrade reqd Headers.empty (fun () ->
+     upgraded := true)
+  in
+  let t = create ~error_handler upgrade_handler in
+  read_request
+    t
+    (Request.create
+      ~headers:(Headers.of_list [ "host", "example.com:80" ])
+      `CONNECT
+      "/");
+  write_response ~msg:"Upgrade response written" t (Response.create `Switching_protocols);
+  Alcotest.(check bool) "Callback was called" true !upgraded;
+  reader_yielded t;
+;;
+
 let tests =
   [ "initial reader state"  , `Quick, test_initial_reader_state
   ; "shutdown reader closed", `Quick, test_reader_is_closed_after_eof
@@ -2520,4 +2538,5 @@ let tests =
   ; "can read more requests after write eof", `Quick, test_can_read_more_requests_after_write_eof
   ; "can read more requests after write eof (before response sent)", `Quick, test_can_read_more_requests_after_write_eof_before_send_response
   ; "write response after reader EOF", `Quick,test_write_response_after_read_eof
+  ; "CONNECT method", `Quick, test_connect_method
   ]