By 610 AD the Empire (Eastern) came under Greek dominance and became what is now called the “Byzantine Empire” – but the citizens still called it “the Land of the Romans.”
As the Western Roman Empire declined during the 5th century, the richer Eastern Roman Empire under Emperor Justinian I would undertake an attempt to regain the majesty of the Roman Empire.   He was successful to a degree as he regained Italy and parts of Illyria from the Ostogoths, North Africa from the Vandals and southern Hispania from the Visigoths.  The reconquest of South Hispania was somewhat ephemeral, but North Africa served the Byzantines for another century, Italy for another 5 centuries and Illyria almost a millennium. 

By contrast, the Eastern Empire would suffer a fate similar to the Western Empire, but at a slower, more convoluted fashion and rate.  Justinian, motivated by a passion to recapture the majesty of the Great Empire, briefly conquered North Africa and Italy, but Byzantine possessions in the West were reduced to south Italy and Sicily.  In the East, the Byzantines were endangered by the expansion of the Islamic tribes who had conquered Syria and Egypt and were threatening Constantinople.  The Byzantines managed to stop the Islamic sweep and expansion during the 8th century.

The Eastern Roman Empire was at its height in 1000 AD, but was defeated in the battle of Manzikert in 1071 AD.  With centuries of internal strife, invasions by the Ottoman-Turks, the sack of Constantinople at the hands of the 4th Crusade (1204), the capture of Constantinople and Constantinople’s recapture by Mehmed II (1453), for many historians, brought the End of the Eastern Roman Empire at 1204 AD. 

The sack of Constantinople at the hands of the 4th Crusade in 1204 AD is sometimes used to mark the end of the Eastern Roman Empire, but the people continued to call themselves Romans until their fall to the Ottoman Turks in 1453 AD with Constantine XI, the last Roman Emperor, which has become The Official End of the Imperial Roman Empire.