Abstract: Stability patterns for the homology of configuration spaces has been studied for decades, starting with homological stability in the 1970's, then representation stability in 2010, and stable periodicity in 2015. When the configuration spaces are Eilenberg-MacLane spaces, these stability results extend to the group homology of surface braid groups. However, configuration spaces on S^2 and RP^2 are not Eilenberg-MacLane spaces and so the stability patterns for their group homology had not yet been established. In this talk, I will use a result from Fred Cohen and Jonathan Pakianathan as well as the theory of FI-homology to establish stability patterns for the group homology of braid groups on S^2 and RP^2.