The Eastern Roman Empire fell in 1453 when the Seljuk Turks conquered Constantinople. The Ottoman Emperor's titles included Kaysar-i-Rum. I can't remember the dates but when the Greeks were conquering Crete their co-linguists described the conquerors as Hellenes while they were Romans.

If you go by the criterion of (primary) self-identification there were Romans until the 1800s at least. By unbroken institutional continuity with the bureaucracy of Empire you can argue over whether it ended in 1453 or 1922.