Rome is said to have fallen in A.D. 476, but this is a simplification. You could say it lasted until A.D. 1453, when the Ottoman Turks conquered the Eastern Roman or Byzantine Empire.
Constantine had set a new capital for the Roman Empire in the Greek-speaking area of Constantinople, in 330. When Odoacer seized Rome in 476, he did not destroy the Roman Empire in the East -- what we now call the Byzantine Empire. The people there might speak Greek or Latin. They were citizens of the Roman Empire.
Even though the western Roman territory was divided into various kingdoms at the end of the fifth and beginning of the the sixth century, the idea of the old, united Roman Empire wasn't lost. Emperor Justinian (r.527-565) is the last of the Byzantine emperors to try reconquering the West.
By the time of the Byzantine Empire, the emperor wore insignia of eastern monarchs, a diadem or crown. He also wore an imperial cloak (chlamys) and people prostrated themselves before him. He was nothing like the original emperor, the princeps, a "first among equals". The bureaucrats and court set a buffer between the emperor and the ordinary people.
Members of the Roman Empire who lived in the East considered themselves Romans, although their culture was more Greek than Roman. This is an important point to remember even when talking about the residents of mainland Greece during the roughly thousand years of the Byzantine Empire.
Although we discuss Byzantine history and the Byzantine Empire, this is a name that was not in use by the people living in Byzantium. As mentioned, they thought they were Romans. The name Byzantine for them was invented in the 18th century.
Photo: Legend-based painting of Belisarius as a Beggar, by François-André Vincent, 1776. Public Domain. Courtesy of Wikipedia.