This is slightly off topic, but I seem to remember reading about some other civilization that saw itself as "a continuation of the Roman Empire with no discernable breaking-off-point in administration, political theory, judicial practice, ruling imperium, economics or culture" and even went so far as to keep calling themselves "Romans" all the way up to the, oh, mid-1400s or something like that ... but darned if I can't remember what we call those guys. Maybe it was the "Dark Empire" or something like that.
The point, as was pointed out, was somewhat overly-obscured in my attempt to be facetious (I'll keep working up to my career as a satirist), but nonetheless one that is important not only for historians, but for those who read the “canon” of Western Great books and care about the story of Western Culture in general—and I hope that covers a large percentage of the avid readership of this blog. On the one hand, the story of Rome’s fall will always be important because of the continuous power of the idea in the life and thought of the West—one can think of Augustine’s City of God (c.413-426) and Gibbon’s Decline and Fall of the Roman Empire (c. 1776-88) as landmarks on each end of the development and use of the story.
On the other hand, scholars who work on the Byzantine Empire and Late Antiquity have done much work in the last decades (Peter Brown, currently at Princeton, is still the most important person to read on this topic and is quite readable) to show that however powerful and influential this story of the Fall of Rome, as far as understanding the life and times, and especially the political and economic structures, and the worldview and culture of citizens of the Roman Empire around the Mediterranean in the 5th and 6th centuries, it is downright false, and at best unhelpful when presented alone and as fact: my own stance is that it is helpful—necessary!—to present the truths present in each of the stories(Rome fell vs. Rome continued), but all of that is for another post.
Today I came across a paragraph by a well-known Patristics scholar at Univ. of Durham, Andrew Louth, that succinctly and cogently explains the view that has come to be accepted by most scholars, and that should be read at least twice at breakfast each day until it is internalized:
The beginning of the [sixth] century saw Anastasius (491-518) on the imperial throne, ruling an empire that was still thought of as essentially the Roman Empire, coextensive with the world of the Mediterranean, however unrealistic such a view seems to modern historians, who have the benefit of hindsight. Although Anastasius ruled from Constantinople, ‘New Rome,’ over what we call the ‘Eastern Empire’, the Western Empire having been carved up into the ‘barbarian kingdoms’, this perspective is ours, not theirs. Through the conferring of titles in the gift of the emperor, and the purchasing of alliances with the wealth of the Empire—wealth that was to dwarf the monetary resources of the West for centuries to come—the barbarian kings could be regarded as client kings, each acknowledging the suzerainty of the emperor in New Rome, and indeed the barbarian kings were frequently happy to regard themselves in this light. The discontinuation of the series of emperors in the West, with the deposition of Romulus Augstulus in 476, was regarded by very few contemporaries as a significant event: the notion that East and West should each have its own emperor was barely of a century’s standing, and the reality of barbarian military power in the West, manipulated from Constantinople, continued, unaffected by the loss of an ‘emperor’ based in the West.
 Note that just about every Medieval survey course you will find starts in 476 with “The Fall of the Roman Empire”. It's not that there aren’t respectable books published by very restectable institutions that still argue for the "old view"—see Bryan Ward-Perkins’ The Fall of Rome and the End of Civilization (2006) ); they just happen to be flat wrong. The caricature of the prevailing view cited in Ward-Perkins’ synopsis, that “there [was] no crisis at all, but simply a peaceful blending of barbarians into Roman culture, an essentially positive transformation” is a straw man argument which no one is saying. Look for a (brief and mostly rant-free, I promise!) review of this book later in the summer in which I will try to focus on why the current orthodoxy is important to get across to anyone who reads from or teaches the Canon of Great Books.
 New Cambridge Medieval History v. I: 500-700, p. 93