The eastern continuation of the Roman Empire — centered on Constantinople, Christian in religion, Greek in language and culture, Roman in legal tradition and political identity — that survived the fall of the Western Empire by a thousand years, preserved and transmitted classical learning, shaped the civilizations of Eastern Europe and Russia, and fell to the Ottoman Turks in 1453, an event that shook the Christian world and sent Greek scholars westward to fuel the Renaissance.
The Byzantines never called themselves Byzantines. They called themselves Romans. They considered themselves the direct continuation of the Roman Empire — which, from one perspective, they were. Their empire lasted from Constantine's founding of Constantinople in 330 CE to its fall in 1453: over eleven hundred years. That is longer than the distance between us and the Norman Conquest. We call it Byzantine — after Byzantium, the Greek city on which Constantinople was built — but that name was invented by Western scholars who needed a way to distinguish the 'real' Rome from its Greek-speaking eastern continuation.