No such entity, of course, lasts for ever unchanged, or indeed at all. Perhaps, indeed, the Roman Empire's first major change was to recognise itself as being ruled by a single, unique Emperor, and that did not, of course, prevent periods of civil war when contending individuals each claimed to be the one Emperor, and then gradually more common periods in which emperors more or less willingly recognised themselves as being co-emperors, still over a single Empire though perhaps with divided responsibilities for different sections of it - until after one such period, the separate imperial authority in one of the sections lapsed, nominally in favour of the one remaining Emperor but with most of the lapsed section in practice being permanently lost to the Empire and the remaining parts only regained for limited periods.
The remaining Roman Empire (and, usually at least in principle, single Emperor) lasted, through various vicissitudes and some discontinuities, for nearly another thousand years in Constantinople (also usually). A few centuries into this period, a rival claimant to the imperial dignity appeared in western Europe - Charlemagne - and this claim generated a series of successors (and separate Empire) which itself continued for just over a thousand years.
But for quite a while longer, there were still rulers recognised as emperors, ruling states internationally recognised as empires, with (rather less generally recognised) claims - or origin myths - to being the true successor of the Roman Empire. Up until 1917, there were four such emperors, with four empires. From 1922 (or 1924, very tendentiously), there were no such emperors - though one of the empires continued without change to its native nomenclature until 1945.
The first to go went in 1917 - the Russian Empire, the self-proclaimed "third Rome", with no territorial or organisational continuity from the previous Romes, but by far the strongest state still following the Byzantine, Orthodox Christian form of Christianity, and in this regard the implicit successor of the Roman Emperors in Constantinople. The Emperor was deposed in February/March 1917, and the Empire officially replaced by a republic that September.
The second and third emperors went in November 1918. The second (by two days) was the German Emperor, the latest established claimant, proclaimed in 1871 as the ruler of an Empire which was the successor, at a couple of removes through intervening confederations of German states, of the Holy Roman Empire which had been dissolved in 1806. The various imperial institutions and the Empire's constituent states went through a very rapid conversion to fully republican governance and personnel, but the state and its central institutions generally kept their old titles until 1945 (but note that the German word Reich does not automatically translate as Empire - in German, France is Frankreich and the United Kingdom the Vereinigtes Koenigreich).
The third was the Emperor of Austria, an empire proclaimed shortly before the dissolution of the Holy Roman Empire by the last Holy Roman Emperor, and since then ruled by him and his descendants. The empire had disintegrated over the previous few weeks, and the remaining German-speaking part of Austria promptly converted itself into a republic.
The fourth and last of these emperors was not deposed until November 1922. While the Ottoman rulers had prioritised their other titles as Sultan and Caliph, they had at least since their conquest of Constantinople in 1453 (and of the remaining fragments of the Byzantine Empire within the next few years) also claimed to be Emperors of Rome (kaisar-i-Rum). The Caliphate remained as a religious title until March 1924 - but presumably the claim of succession to the Roman Empire disappeared with the Sultanate.
None of these imperial titles have ever shown any serious signs of revival - or, after a hundred years, any signs at all of doing so in future. But their disappearance changed the political face of Europe and the world - in some ways for the better, but in other ways probably for the worse. I have some more thoughts on this, but they are matters for a further post.