To me this is important, because you can't be a Roman Empire without being Roman, or rather, considering yourself Roman. I want to know. I know for sure that the Greek Byzantines considered themselves Roman, but I know of no others. I'd prefer to know of places where both the rulers and the populace considered themselves Roman, as I don't think the populace of the Holy Roman Empire considered themselves Roman.
I don't think I'm sure what exactly you're asking - do you want to know who considered themselves Roman after the fall of the western Roman Empire? Or who considered themselves Roman during antiquity?