Personally i think the Roman empire began dying when the old gods were swept aside and Christianity became the official religion, and then finally died in 476AD when the last emperor was deposed by Odocer(sp).
I don't think the eastern Roman empire can still be considered a continuation of Rome simply because the western empire had dissolved and the city of Rome was ruled by barbarians with no senate and no powers in the hands of the old Romans. After that the eastern empire was just an empire that had broken off from the old Roman empire.
Also the Holy Roman Empire is a joke, everybody knows the old saying... Neither holy, nor Roman, nor an empire. Other peoples tried to be the new Rome but without the city and the actual Roman people that's just not possible.
What's everybody elses opinion on it? Do you think the Roman empire died when the Byzantine empire died, etc. etc? If so, why?