With given places ramifying, which groups can occur as Galois groups over a field K? For function fields K of curves over algebraically closed fields k of characteristic p, this is answered by Abyhankar's Conjecture, which was proved by M. Raynaud and D. Harbater. For number fields K, I will generalize some results in Harbater's papers to give some answers to the problem above, especially when there is just one ramified prime. For function fields K over finite fields, I will give some constraints on the groups that can occur with given ramification, and take steps to develop Iwasawa theory for cyclotomic function fields.