It is widely believed in the West that Rome fell in 476 CE, when the Germanic chieftain Odoacer deposed the boy-emperor Romulus Augustulus. This is, technically, half true. The Western Roman Empire fell in 476. The Eastern Roman Empire — administered from Constantinople, speaking Greek but calling itself Roman until the day it fell — continued for another nine hundred and seventy-seven years, until the Ottoman Turks took Constantinople in 1453. Its sixth-century emperor Justinian even briefly reconquered Italy, North Africa, and southern Spain, so that for a moment the Mediterranean was again a Roman lake. The Byzantines did not call themselves Byzantines — that label was coined by a German historian a century after the empire was gone. They called themselves Romaioi, Romans. They were not wrong.
Byzantium spent a thousand years preserving the administrative, legal, and intellectual continuity of the Roman Empire while the Latin West worked through its own much shorter dark age. The Justinian Code (529–534 CE) compiled centuries of Roman law into a corpus that, rediscovered in eleventh-century Bologna, became the foundation of every later civil-law system in continental Europe and Latin America. Byzantine missionaries — the brothers Cyril and Methodius — Christianized the Slavs, gave them the Cyrillic alphabet, and shaped what became Orthodox Christianity from Kiev to Belgrade. Constantinople was, for most of its run, the largest and richest city in the Christian world, its triple Theodosian walls unbreached for a thousand years, a hinge between Mediterranean Christianity and Islam and the terminus of the Silk Road. The empire shrank, recovered under the Macedonian dynasty, and shrank again after the catastrophe at Manzikert in 1071 opened Anatolia to the Turks. The Fourth Crusade's sack of the city in 1204 — by fellow Christians — was the wound it never fully healed: even after the Byzantines retook Constantinople in 1261, it left a rump state the Ottomans finally extinguished in 1453. That fall is the canonical date for the end of the Middle Ages, and the event that drove Greek scholars west into Italy with their manuscripts — helping ignite the Renaissance.
Modern Russia explicitly understands itself as the Third Rome — heir to the Byzantine inheritance after Constantinople's fall, a doctrine first articulated by the monk Philotheus around 1510 — and the Putin regime invokes this geopolitical theology when convenient, casting Moscow as defender of an Orthodox civilization against a decadent West. The Byzantine continuity is also why the Orthodox–Catholic schism, dating to 1054, still maps roughly onto the Russia-versus-West fault line a millennium later, visible in the very church politics surrounding Ukraine.