This question is asked pretty frequently; one of the more comprehensive answers (by u/Astrogator) can be found here

I read in the mentioned u/Astrogator response:

The usual process by which newborn Romans were registered into the citizen body is detailed by Sherwin-White: "The Roman citizen was required to register the birth of his children within thirty days before a Roman official, and he received a wooden diptych recording the declaration, which acted as a certificate of citizenship for the child for the rest of his life. Like the military diplomata this contained the names of seven witnesses, and provided a presumptive proof of citizen status."9

What happened if a Roman citizen gave birth to a child outside Roman territory or inside Roman territory but in a place where there were no Romans officials nearby to register the birth. How could the child's citizenship be proven well after its birth? Was a child born outside Roman territory to Roman citizens considered a Roman citizen automatically even if he had not set foot on Romer for years? (Say until adulthood?)

Also what happened with children born inside Roman territory to non-Romans? Was there some sort of "naturalisation" process?

Finally could someone acquire Roman citizenship later in life by some standardized process, other than by some special circumstances decree?