Given a Galois branched cover of the projective line in characteristic p, we can ask whether it lifts to a Galois branched cover of the p-adic line.  Conversely, given such a cover of the p-adic line, we can ask whether it is a lift of a characteristic p cover, and if so, what can be said about that cover.  In this talk, I will especially focus on the case that the Galois group is Z/pZ, surveying known results, and talking about my current research directions.