Suppose given some object over a field of characteristic p (think of an algebraic curve, an elliptic curve with an endomorphism, an algebraic surface, a finite group scheme, a curve with an automorphism, a CM abelian variety). Can we lift this to characteristic zero?

General methods (local moduli and algebraization) show this can easily be done in some cases. In the last 50 years we have seen solutions of this problem in many more cases. We give a survey.

In some situations obstructions of various kinds prevent a positive answer.

We show various methods and ideas involved. We see that in almost all situations a combination of standard techniques plus new ideas and constructions seem necessary to obtain an answer. We will discuss the "Mumford method". We describe classical approaches and recent spectacular results. We give proofs, general ideas and specific examples.