So tsar and kaiser both mean Caesar. Charlemagne's empire was pointedly labelled a Roman one. Mehmed II claimed to be succesor to the Romans after his conquest of Constantinople in 1453. Were any of these claims, or others, legitimate? Is the idea of "succesor states" at all realistic?
The thing is, whether such a claim is legitimate, is completely open to interpretation! Chalemagne reigned over much of the old roman territory, was hailed as emperor in the city of Rome, and one could argue that the Carolingian Rennaissance was a return to old roman literacy. But it would be just as right to argue the complete opposite. After all, his empire was vastly different in almost every conceivable way - and it had been more the 300 years since the last roman emperor - how can you truly call it a successor state after so long?
Mehmeds claim is of a different nature, he marched into Constantinople, defeated the last (more or less) continous remains of Rome in the form of the Byzantines and thus claimed descent as ruler of the city, and the empire. In a lot of ways the Ottoman Empire was very alike to the conquered Byzantine, and is thus - to some - a more, if not the most, 'direct successor' to Rome - The Holy Roman Empire of the time was even more unalike to both Byzantium and the ancient Rome.
But then, Ivan III married a niece of the last byzantine emperor in Moscow. In a dynastic view, this made Russia the 'Third Rome' - but dynasticism was very different in ancient Rome, and I doubt his wife Sophia could have claimed descent from Augustus or the other ancient emperors. But as (one of) the last remaining Orthodox nations, he could claim leadership as the heir of Constantinople, and in turn of Rome, especially in a christian sense, as these cities had been the centres of early christianity, and the emperors after Constantine had played quite a role in the early church.
So, in short, they were all legitimate to the people making the claims and in the end, isn't that mostly what matters? It's a difficult thing to sit here, so many centuries later, and defining who had the better claim. Personally, I think that whatever we say about it as historians today is of no consequence. The people within the Byzantine Empire called themselves Roman until the very end(and quite a bit longer) - so why shouldn't we treat them as such, at least until there is a serious break one cannot overlook?