The end of the Roman Empire is traditionally placed on 4 September 476, as the Western Roman Empire fell to Germanic invaders.
However, this view does not recognize the Eastern Roman Empire, known to modern-day historians as the Byzantine Empire, which maintained Roman legal and cultural traditions. Developing a distinct Greek Christian character, it managed to survive and even thrive for a millennium after the fall of the West, eventually being conquered on 29 May 1453 by the Ottoman Empire.
There are many reasons speculated as to why this was so, but mainly problems associated with internal unrest, resources spread too far and wide and external forces gaining power and territory. More detail here:
http://en.wikipedia.org/wiki/Decline_of_the_Ro man_Empire