From Wikipedia
The decline of the Roman Empire refers to both the gradual disintegration of the economy of Rome and the barbarian invasions that were its final doom. The English historian Edward Gibbon, author of The Decline and Fall of the Roman Empire (1776) made this concept part of the framework of the English language, but he was neither the first nor the last to speculate on why and when the Empire collapsed. "From the eighteenth century onward," Glen W. Bowersock has remarked, "we have been obsessed with the fall: it has been valued as an archetype for every perceived decline, and, hence, as a symbol for our own fears." It remains one of the greatest historical questions, and has a tradition rich in scholarly interest. In 1984, German professor Alexander Demandt published a collection of 210 theories on why Rome fell, and new theories have emerged since then.
This slow decline occurred over an estimated period of 320 years which many historians believe finally culminated on September 4, 476 when Romulus Augustus, the last Emperor of the Western Roman Empire was deposed by Odoacer, the Visigoth king. To an extent any such date must be arbitrary; Julius Nepos, the legitimate emperor recognized by the East Roman Empire continued to live in Salona, Dalmatia until he was assassinated in 480. Some modern historians question the relevance of this date, as the Ostrogoths who succeeded considered themselves as upholders of the direct line of Roman traditions, and noting, as Gibbon did, that the Eastern Roman Empire was going from strength to strength and continued until the Fall of Constantinople on May 29, 1453. Some other notable dates are the Battle of Adrianople in 378, the death of Theodosius I in 395 (the last time the Roman Empire was politically unified), the crossing of the Rhine in 406 by Germanic tribes after the withdrawal of the troops in order to defend Italy against Alaric I (such invasions had occurred many times previously but this time it was successful), the death of Stilicho in 408, followed by the disintegration of the western army, the Sack of Rome (410), the first time in almost 800 years that the city of Rome had fallen to a foreign enemy, the death of Justinian I, the last Roman Emperor who tried to reconquer the west, in 565, and the coming of Islam after 632. Many scholars maintain that rather than a "fall", the changes can more accurately be described as a complex transformation. Over time many theories have been proposed on why the Empire fell, or whether indeed it fell at all.

The Western and Eastern Roman Empires by 476