I sent a registered letter but they claim it never arrived. What do you do?

With a registered letter - you should have been given a receipt from the post office when you paid for the registered service. That receipt has a unique number. The post office - when they delivered the letter, will have a signature of the person who accepted it from the postal worker, matched against their delivery register. If the post office confirm they delivered it (and provide you with proof) - you simply contact the place you sent the letter to - and inform them you can prove they received it !