Ottoman Turks under the command of Mehmed II break into Constantinople and capture the city. With this, the last remnant of the Roman Empire is destroyed and Constantine XI Palaeologus, the last Byzantine emperor, dies.
By this point there's little left of the empire, just the city of Constantinople and some land around it in the Greek province of Thrace. Both the culture and the language had long since become Greek rather than Roman.
The Ottomans, however, consider themselves to be the legitimate successors of the Byzantine emperors and commonly use the title Sultan-i Rum, Sultan of Rome.