When you have a connection issue, make sure to first check Telegram's Twitter account. The Telegram team will announce widespread problems on Twitter.
If the problem is local only, Telegram needs to know some information before they can investigate the issue.
- Do other (non-Telegram) apps work fine for you?
- What exactly is happening? (does it say 'Connecting...' / 'Updating...' all the time? Are messages not sent / delivered?)
- When exactly did it start and is it still happening?
- Does this happen with all Telegram apps or just one?
- Does Telegram's web version work on your device?
- Does the issue only happen when you are connected via WiFi or also on a mobile connection or maybe even in both cases?
- In case you can successfully connect, try out Telegram's connectivity test channel. Which GIFs are downloaded slowly or not downloaded at all?
It may be possible that Telegram needs a traceroute to their servers later. But before doing so, wait for instructions from Telegram Support to get the right IP to trace the route to — each case potentially requires a different one.
If you are having issues receiving the SMS login code, contact email@example.com. Send them the name of your mobile operator and your phone number.
Please only do this after Telegram Support asked for a traceroute and gave you the IP address of the specific server.
Traceroute on iOS
A lot of traceroute apps exist in the AppStore, unfortunately a lot seem to be broken. A working one is called Manage Engine Ping Tool. You can download their free app here: https://apps.apple.com/app/manageengine-ping-tool/id460362949.
- After installation, start the app (notification permission is not needed)
- Select Traceroute in the menu
- Enter the IP address 
- Press on TRACE 
- After a minute or two, the traceroute should have been completed, then tap on COPY 
- Again top right, this time OPEN WITH and a PDF will be created with the complete route.
- Share the PDF with the support team
Traceroute on Android
Same issue as on iOS. A lot of non-working Traceroute apps. A working one is called he.net - Network Tools. You can download their free app here: https://play.google.com/store/apps/details?id=net.he.networktools.
- After installation, start the app
- Select the top left button  to open the sidebar on the left
- Select TRACEROUTE, enter the IP address  and press ENTER on the keyboard
- After a minute or two, the traceroute should have been completed. Tap on '...'  top right
- Select SHARE and send it to the support team