Upvote:2
Well, the Byzantine Empire certainly considered itself to be a continuation of the Roman Empire. It continued Roman culture fairly directly at first, but gradually changed over its lifespan of nearly a thousand years. The Empire had been re-arranged and sub-divided a number of times, but the distinction between the Greek-speaking East and Latin-speaking West was the one that lasted, and the Byzantine Empire had started as the Eastern Roman Empire.
The Holy Roman Empire also considered itself to be a continuation of the Roman Empire. It only maintained Latin as a language of the educated after the Romance languages had developed out of Vulgar Latin, and most of the HRE territory was German-speaking anyway. While its culture had Roman influences, it wasn't strictly Roman. It did, however, help preserve Latin Christianity.
The Ottoman Empire regarded itself as a successor to the Byzantine Empire, but it wasn't founded until about 800 years after the fall of the Western Roman Empire, so the claim to be a successor of the original Rome was not taken seriously, and lapsed after the death of Mehmed II.
The Russian claim was based on family links to the last Byzantine emperors, and their support of Orthodox Christianity. "Tsar" is their version of "Cesar".
After the dismantling of the Holy Roman Empire, the Austrian Empire claimed to be its successor, as did the German Empire from 1871. Fascist Italy also claimed to be a new Rome, but they weren't very good at conquest.
The Wikipedia article on Third Rome is a reasonable summary of the various claims.