Are there any descendents of the Byzantine Emperors still alive?

Question

I always wonder what happened to them all after the turks took Constantinople

Answer 1

The Emperors children intermarried with other European royal families, particularly in Russia, which shared Orthodox religion. The Tsars later used this to declare themselves the Third Rome. This sentiment began during the reign of Ivan III, Grand Duke of Moscow who had married Sophia Paleologue. Sophia was a niece of Constantine XI, the last Eastern Roman Emperor and Ivan could claim to be the heir of the fallen Eastern Roman Empire.

Answer 2

The last emperor, Constantine XI, did not have children before he died in the fall of Constantinople, so there are no direct descendents. His brothers' male children did not survive to reproduce. There were lots of cousins, though, who survived, married and procreated, so there are many descendents of the family, none of the actual emperors.

Answer 3

According to Bishops at Large by Peter Anson there is a gentleman living in Rome who claims to be the heir to the Byzantine throne and styles himself Basileus of the Byzantine Empire. He's even managed to get some oddball Orthodox Greek Catholic prelate (yes, there is such a thing) to consecrate him as such.