Let X be a smooth curve over a field of characteristic p>0 and G a finite group of automorphisms of X. The "lifting problem" is concerned with finding a G-equivariant lift of X to characteristic zero. This problem can be reduced to the study of finite automorphism groups of a p-adic unit disk. We discuss several recent results on this problems. Some of them are positive, i.e. solve the problem in special cases, others are negative, i.e. yield obstructions against liftability.