The rulers and citizens of the Byzantine Empire always viewed themselves as Romans, and in fact the name Byzantine was only applied to them after the fact, by historians.
But what did the other kingdoms of the ages call the empire? Were they known as the Roman Empire even by medieval Europe? Were they recognized by the rest of the world as the continuation of Rome they believed themselves to be? Or were they simply seen as Greeks (the language mainly spoken there instead of Latin) or some other nationality?
The 'Byzantines' themselves referred to themselves as the "Greek: Βασιλεία τῶν Ῥωμαίων, tr. Basileia tôn Rhōmaiōn; Latin: Imperium Romanum". The other major powers at the time, namely the Muslim Caliphates such as the Abyssids etc, also saw it as the Roman Empire (there is a sura in the Qur'an called "Surat Ar-Rum" which is the sura dealing with the Romans, translated today sometimes as the sura dealing with the Byzantines).
Before the year 799, the Western world also universally acknowledged the Eastern Roman Empire as the true Roman Empire. However after 799, the Bishop of Rome used the excuse that since the Empress Irene usurped her son to become the ruler of the ERE, the throne of the Roman Emperor was empty (as the idea that a woman could be the Emperor of Rome was ridiculous according to western culture at the time). Thus he was able to crown Charlemagne on December 25th, 800 C.E with that title, and also create his position as the most important figure in all of Christendom, a.k.a the Pope (also interesting to note that Christmas was not celebrated on exactly Dec 25th until afterwards; previously it was celebrated around that day, but not in an exact way. The date itself is derived from an old Roman holiday to celebrate the cult of the sun, established by Marcus Aurelius.).
From then on, the west (in this case the HRE and the papacy, France was usually far more diplomatic, as the HRE was a potential threat to it, even if it was inward looking) generally did not like to give the Eastern Roman Empire the legitimacy that they themselves had claimed. If they were being diplomatic, they would call them the Romanians (a title that the Eastern Romans would sometimes use to call themselves), while the 'Holy Roman Empire' would be the Imperium Romanum. If they wanted to give a diplomatic snub to the empire, they would then call them the Imperium Graecorum, the Empire of the Greeks. Of course, the Bulgarians, Avar Khaganate and other powers in the Balkans also recognized the Eastern Romans as the Empire of the Romans (the Bulgarians did occasionally claim the title themselves after a major victory in a campaign, which they sometimes had.).