I will begin by giving a brief introduction to the general problem of lifting Galois covers of curves from characteristic p to characteristic zero, before narrowing my focus to the case of p^n-cyclic covers. In this context, I will describe work of Green and Matignon that solves the lifting problem positively in the case of p and p^2-cyclic covers. In addition, by studying the geometry of order p automorphisms of the open p-adic disc, they have deduced local obstructions to the liftability of noncyclic abelian covers.