We prove there is no ring with unit group isomorphic to Sn for n ≥ 5 and that there is no ring with unit group isomorphic to An for n ≥ 5, n \neq 8. To prove the non-existence of such a ring, we prove the non-existence of a certain ideal in the group algebra F_2[G], with G an alternating or symmetric group as above. We also give examples of rings with unit groups isomorphic to S1, S2, S3, S4, A1, A2, A3, A4, and A8. Most of our existence results are well-known, and we recall them only briefly; however, we expect the construction of a ring with unit group isomorphic to S4 to be new, and so we treat it in detail.