The Visigoths (410 AD) and the Vandals (455 AD) sacked Rome, then the Ostrogoths deposed the last emperor of the west (476 AD). The Persians never seriously threatened Rome after Trajan invaded Parthia and seized the capital Ctesiphon (116 AD).
As for Constantinople, it was sacked only twice: by the 4th Crusade in 1204 and by the Ottoman Turks in 1453.
Just got done reading a pretty good treatment of the 4th crusade by Jonathan Phillips. Have you seen that one?