When the Roman Empire fell, it did not fall as a whole simultaneously - Byzantium carried the torch while order changed and empires regrouped. The empire of Rome in the west had fallen to the invaders from the Germanic regions but in the east the Byzantine part of the empire held up. Over a thousand years it stood as a fortress against Arab expansion.
The Byzantine Empire kept the fires of the previous civilization burning - and added new innovations. The Greek language, research and attitude to learning were protected whilst the Romans imperial administration was kept and Roman laws enshrined. The religion of Greek Orthodox worship converted many new immigrants to an alternative form of Christianity glorifying it through its new religious art. Constantinople was in a fulchrum position between east and west becoming the conduit of culture for new migrants. Byzantium's 'capital' was diverse, prosperous and vibrant and influenced other influential civilizations of the Middle Ages. Up until the the empire fell in 1453, Byzantium's cultural influence, religious ambitions and political ideas had produced fruitful results among the eastern Europe peoples including in Russia - indeed the Russians eventually claimed the legacy of Byzantium.