The modern western world could not have existed without the Byzantine empire, which influenced the Muslim world and western Europe throughout the Middle Ages.
The influence waxed and waned with the fate of the empire and its many enemies, but was a constant factor.
The grandeur of Constantinople was reflected by the imperial court, the diplomatic service and bureaucracy, the ceremony of coronation (copied by western monarchs, first by the German emperors), the legal system and adequate administration and the deeply rooted culture that sprang from ancient, Christian, Jewish and later islamic sources.
The imperial identity was strengthened by a linguistic continuity that linked medieval scholars back to Greek culture and to encourage them to preserve ancient texts.
Byzantium combined the ancient heritage with the Christian belief. The inheritance of the Roman technology and engineering skills guaranteed the construction of aqueducts, fortifications, roads, bridges and huge constructions, which even surpass the accomplishments of architects in Rome.
The St. Sophia from the sixth century is the largest dome ever built until the construction of the St. Peter thousand years later. The Roman legal system and military tradition were eagerly adopted.
Although Byzantium controlled a much smaller empire than Rome at its height, the empire developed new political and cultural forms and it halted the Muslim expansion in 678, allowing the fragmented western European sovereignties to develop their own strengths.
Western Europe was hopelessly fragmented, politically and linguistically, and the Pope was not in a position to unite the Christian world.
Byzantium is still everywhere, in museums, churches, cellars, books and many other works of art and (public) buildings in Russia, Spain, Turkey, Greece, Cyprus, Germany, England, Ireland and other countries.
Byzantium played a crucial role between the East and the West for many centuries and the current accession talks with Turkey might be put into this perspective. Source: J. Herrin, Byzantium. The Surprising life of a Medieval Empire (London 2007).