It is accepted that the Byzantine Empire fell in 1453 to the Ottomans under Mehmed II finally took over Constantinople after 53 days of battle. However, there were several other successor states to the Byzantine Empire such as the Despotate of Morea and the Empire of Trebizond that could be argued to be "Byzantine". Some didn't even fall until 1479. That being said, the Roman "Byzantine" Empire definitely fell in 1453.