diff --git a/lib/bandit/pipeline.ex b/lib/bandit/pipeline.ex
index 0bc51f87..71c01c93 100644
--- a/lib/bandit/pipeline.ex
+++ b/lib/bandit/pipeline.ex
@@ -49,11 +49,7 @@ defmodule Bandit.Pipeline do
         exception ->
           handle_error(:error, exception, __STACKTRACE__, transport, span, opts, conn)
       catch
-        :throw, value ->
-          handle_error(:throw, value, __STACKTRACE__, transport, span, opts, conn)
-
-        :exit, value ->
-          handle_error(:exit, value, __STACKTRACE__, transport, span, opts, conn)
+        kind, value -> handle_error(kind, value, __STACKTRACE__, transport, span, opts, conn)
       end
     rescue
       exception ->
diff --git a/lib/bandit/telemetry.ex b/lib/bandit/telemetry.ex
index 51e9e8aa..0fdfa805 100644
--- a/lib/bandit/telemetry.ex
+++ b/lib/bandit/telemetry.ex
@@ -77,7 +77,8 @@ defmodule Bandit.Telemetry do
         is also set and the nature of error is such that Bandit was unable to successfully build
         the conn
       * `kind`: The kind of unexpected condition, typically `:exit`
-      * `exception`: The exception which caused this unexpected termination
+      * `exception`: The exception which caused this unexpected termination. May be an exception
+        or an arbitrary value when the event was an uncaught throw or an exit
       * `stacktrace`: The stacktrace of the location which caused this unexpected termination
 
   ## `[:bandit, :websocket, *]`
@@ -190,11 +191,13 @@ defmodule Bandit.Telemetry do
 
   @spec span_exception(t(), Exception.kind(), Exception.t() | term(), Exception.stacktrace()) ::
           :ok
-  def span_exception(span, :error, reason, stacktrace) when is_exception(reason) do
+  def span_exception(span, kind, reason, stacktrace) do
+    # Using :exit for backwards-compatibility with Bandit =< 1.5.7
+    kind = if kind == :error, do: :exit, else: kind
+
     metadata =
       Map.merge(span.start_metadata, %{
-        # Using :exit for backwards-compatibility with Bandit =< 1.5.7
-        kind: :exit,
+        kind: kind,
         exception: reason,
         stacktrace: stacktrace
       })
@@ -202,10 +205,6 @@ defmodule Bandit.Telemetry do
     span_event(span, :exception, %{}, metadata)
   end
 
-  def span_exception(_span, _kind, _reason, _stacktrace) do
-    :ok
-  end
-
   @doc false
   @spec span_event(t(), span_name(), :telemetry.event_measurements(), :telemetry.event_metadata()) ::
           :ok
diff --git a/test/bandit/http1/request_test.exs b/test/bandit/http1/request_test.exs
index e8268582..4c2de5a1 100644
--- a/test/bandit/http1/request_test.exs
+++ b/test/bandit/http1/request_test.exs
@@ -2413,7 +2413,7 @@ defmodule HTTP1RequestTest do
       assert output =~ "(RuntimeError) boom"
     end
 
-    test "it should not send `exception` events for throwing requests", context do
+    test "it should send `exception` events for throwing requests", context do
       output =
         capture_log(fn ->
           {:ok, collector_pid} =
@@ -2423,7 +2423,18 @@ defmodule HTTP1RequestTest do
 
           Process.sleep(100)
 
-          assert [] = Bandit.TelemetryCollector.get_events(collector_pid)
+          assert Bandit.TelemetryCollector.get_events(collector_pid)
+                 ~> [
+                   {[:bandit, :request, :exception], %{monotonic_time: integer()},
+                    %{
+                      connection_telemetry_span_context: reference(),
+                      telemetry_span_context: reference(),
+                      conn: struct_like(Plug.Conn, []),
+                      kind: :throw,
+                      exception: "thrown",
+                      stacktrace: list()
+                    }}
+                 ]
         end)
 
       assert output =~ "(throw) \"thrown\""
@@ -2433,7 +2444,7 @@ defmodule HTTP1RequestTest do
       throw("thrown")
     end
 
-    test "it should not send `exception` events for exiting requests", context do
+    test "it should send `exception` events for exiting requests", context do
       output =
         capture_log(fn ->
           {:ok, collector_pid} =
@@ -2443,7 +2454,18 @@ defmodule HTTP1RequestTest do
 
           Process.sleep(100)
 
-          assert [] = Bandit.TelemetryCollector.get_events(collector_pid)
+          assert Bandit.TelemetryCollector.get_events(collector_pid)
+                 ~> [
+                   {[:bandit, :request, :exception], %{monotonic_time: integer()},
+                    %{
+                      connection_telemetry_span_context: reference(),
+                      telemetry_span_context: reference(),
+                      conn: struct_like(Plug.Conn, []),
+                      kind: :exit,
+                      exception: "exited",
+                      stacktrace: list()
+                    }}
+                 ]
         end)
 
       assert output =~ "(exit) \"exited\""
diff --git a/test/bandit/http2/plug_test.exs b/test/bandit/http2/plug_test.exs
index 13cd525d..a18dd346 100644
--- a/test/bandit/http2/plug_test.exs
+++ b/test/bandit/http2/plug_test.exs
@@ -1027,7 +1027,7 @@ defmodule HTTP2PlugTest do
       raise "boom"
     end
 
-    test "it should not send `exception` events for throwing requests", context do
+    test "it should send `exception` events for throwing requests", context do
       output =
         capture_log(fn ->
           {:ok, collector_pid} =
@@ -1037,7 +1037,18 @@ defmodule HTTP2PlugTest do
 
           Process.sleep(100)
 
-          assert [] = Bandit.TelemetryCollector.get_events(collector_pid)
+          assert Bandit.TelemetryCollector.get_events(collector_pid)
+                 ~> [
+                   {[:bandit, :request, :exception], %{monotonic_time: integer()},
+                    %{
+                      connection_telemetry_span_context: reference(),
+                      telemetry_span_context: reference(),
+                      conn: struct_like(Plug.Conn, []),
+                      kind: :throw,
+                      exception: "thrown",
+                      stacktrace: list()
+                    }}
+                 ]
         end)
 
       assert output =~ "(throw) \"thrown\""
@@ -1047,7 +1058,7 @@ defmodule HTTP2PlugTest do
       throw("thrown")
     end
 
-    test "it should not send `exception` events for exiting requests", context do
+    test "it should send `exception` events for exiting requests", context do
       output =
         capture_log(fn ->
           {:ok, collector_pid} =
@@ -1057,7 +1068,18 @@ defmodule HTTP2PlugTest do
 
           Process.sleep(100)
 
-          assert [] = Bandit.TelemetryCollector.get_events(collector_pid)
+          assert Bandit.TelemetryCollector.get_events(collector_pid)
+                 ~> [
+                   {[:bandit, :request, :exception], %{monotonic_time: integer()},
+                    %{
+                      connection_telemetry_span_context: reference(),
+                      telemetry_span_context: reference(),
+                      conn: struct_like(Plug.Conn, []),
+                      kind: :exit,
+                      exception: "exited",
+                      stacktrace: list()
+                    }}
+                 ]
         end)
 
       assert output =~ "(exit) \"exited\""