The Western Roman Empire was conquered in 476, when Germanic tribes led by Odoacer sacked Rome and deposed the Emperor, Romulus Augustulus. The Eastern Roman Empire fell in 1453, when the Turks led by Sultan Mehmed II invaded and claimed control of Constantinople after a 40 day siege.
Barbarian tribes such as the Huns, Goths and Vandals presented an aggressive military presence at Rome's borders, starting in the 4th century. Government corruption, infighting and a army that was thinned due to its huge territory led to Rome's eventual fall to these tribes.
In the Eastern Empire, poor leadership and a loss of military might directly influenced Constantinople's demise.