Are you asking about...

(1) DNA evidence? See these science-y posts, where we see that every living person of European descent must be descended from pretty much every European who lived that long ago.

• How closely related are modern Italians and ancient Romans? - although this discusses only modern Italians, not modern people elsewhere

• AMA from /r/AskScience: AMA: We are the authors of a recent paper on genetic genealogy and relatedness among the people of Europe. Ask us anything about our paper! - not particularly about "Romans"

(2) Genealogical proof? i.e. Are there living people who are able to trace their ancestry back to the Western or Eastern Roman Empire? Apparently not: it seems that genealogical records are too spotty to conclusively connect the dots that far back. Here are a few posts:

• Did any roman patrician families survive the fall of the western empire? Is there anyone alive today that can claim to be a descendant of a roman senator? - includes a comment by u/Georgy_K_Zhukov who demonstrates from a mathematical & logical perspective, that every European alive today must be descended from everyone that long ago

• How much did the Byzantine Empire change between the 6th and 15th centuries?

• What happened to the Patrician bloodlines after the fall of Rome?

• Are there any European families that still exist today that can trace their origins to the aristocracy/patrician families of the Roman Empire?

(3) Inheritors of "Roman" identity? That one's too complex for a quick answer. I leave the floor open to others.