I understand that the Roman empire was a cosmopolitan and multi-ethnic empire, and that as time went on, more and more people in the empire were integrated and granted citizenship. How did the Romans conceptualize race? I'm asking this in the broadest sense -- any information you have on it. Other questions: did they have any conception or prohibition against miscegenation? Was Rome itself open to all peoples across the empire, or strictly for the residents?

I remember reading a while ago a post here or in /r/history about how colonizers viewed race. I can't find it now, but it had to say that one view was that race was emergent of the food that locals vs foreigners ate. I'd like to reference that post, but I can't find it anywhere.