476 is a nice and simple date to remember, but it is patently wrong to say that the Roman empire 'fell' in 476, since er, half of the empire didn't. There are other reasons to consider as well:

• Romulus Augustulus, the 'last Roman emperor', was a barely legitimate usurper and only the more acceptable face of his father's regime, so his reign wasn't really that big a deal.
• A much more legitimate emperor, Julius Nepos, was hanging around Dalmatia. Why should we prioritise a usurper's short reign over someone who had much more right to the throne? He died in 480, which is another date for when the imperial line can be said to have died out.
• Why was Nepos more legitimate? Well, it was because he was backed by the eastern Roman empire, who then tacitly recognised Odoacer's claim that he was ruling Italy in Constantinople's name.
• Odoacer was just another Roman general overthrowing someone he didn't like. People did that all the time, so I don't think 476 was that important from this perspective.
• More importantly, the 'empire' had already lost all its territorial toeholds outside of Italy in the preceding years, it seems to me that the end of the western empire was a gradual one, rather than one that reached its climax in 476.
• The Romans in Italy were also pretty okay with the transition, since Odoacer and his successors generally tried to maintain the political system. They grew up and found success within a Roman framework, so they had little incentive to turn the world upside down. Theodoric the Great of the Ostrogoths is an excellent example of this, since he was acclaimed as an augustus and a princeps by his subjects and ruled in the Roman manner. Why shouldn't we see his kingdom in the sixth century as a renewed Roman empire? Let's not forget Clovis the Frankish augustus from Merovingian Gaul at the same time as well.
• Lastly, I think it's important to recognise that 476 was an artificial construction in terms of its importance, since it was first said to be the 'end' of the empire by the chronicler Marcellinus Comes c.518. Marcellinus was based in Constantinople and wrote at a time when relationship between the empire and Ostrogothic Italy reached an all-time low, so a somewhat polemical treatment of the 'successors' to the empire in Italy to attack the legitimacy of the highly successful Ostrogothic regime was pretty useful. Aside from Marcellinus, 476 was seen as just one of many examples of regime change in this period, no big deal really.

To be honest, I think it's much more interesting to study what happened afterwards. In the early sixth century all the 'barbarian' kingdoms minted coins in the eastern emperor's name, but over time this changed and they started to put their names on their coins, which suggests that there was some sort of a turning point in the mid-sixth century. One reason might be that with the Romans once again ruling Italy after Justinian I's reconquest the various western powers found it necessary to redefine themselves as something other than theoretical subjects of the emperor. This to me seems to be far more important that what happened in 476 :/ Another important period to note is the seventh century, which is when some historians think the eastern Roman empire became the 'Byzantine empire', largely due to the fiscal/military/political crises it faced and the reforms it made to survive it. I don't buy it at all, but again, it's a much more important period of transformation than the coup of 476, which was a far more ephemeral event than many people still think it is.

Edit: Here's something I wrote about the west after 476 more generally, which should give you a sense of what the reaction was elsewhere.

What exactly do you mean by "already dead"? The Eastern half of the Empire thrived for much of the fifth century, it most certainly was not "dead". Rome as a singular political entity did not fall in 476. While the Western half of the Empire does indeed cease to function, it continued to exist for over a thousand years in the East. For much of the medieval period, the Roman Empire was extremely wealthy and powerful, it's culture greatly influencing it's surrounding regions. So, yes, modern scholars generally tend to date the end of the Empire to 1453. 476, while an important date, did not mark the end of the Roman state, and Roman culture and identities continued to exist in the West for centuries.

I think the answer to the question requires another question: what makes the Roman Empire "Roman"? Is it the city of Rome? Is it Roman culture? Is it Roman political institutions? Is it the line of succession? Is it Christianity? All of these have been seriously considered as the defining features of the Roman Empire.

I dislike 476 as the date for the fall (though I do use it when I teach, ugh), but I also think it's disingenuous to use 1453 without an awful lot of explanation. I would even suggest 235 (death of Alexander Severus) or 284 (Diocletian founds the tetrarchy) as alternate dates for the Fall of Rome because the empire that emerges after those dates is significantly different than the one that existed before.

If you think Rome is comparable to Chinese dynasties, then we could view any political shake-up as a fall. 45 BCE: the Rebublic falls. 27 CE: the Empire is founded. 235: the Empire falls. 284: the Tetrarchy is founded. Or you could even have a "fall" at the end of each dynasty.

For real though, I would consider the fall of Rome (though I would argue that it's more a pivot than a fall) to be 629 CE.

In 629, Heraclius declared himself basileus after conquering Persia. He was the first Roman emperor to adopt a title (they signaled their position by taking the name Augustus). This signals a major shift from Latin culture to the Greek culture that better represented the people under the empire.

Just for curiosity's sake, here are other dates that have been seriously suggested:

• 1204: Venetian sack of Constantinople

• 1461: Fall of the last Byzantine army

• 1806: Fall of the Holy Roman Empire under Napolean

• 1917: Death of Tsar Nicholas II