Tate proved that any abelian variety defined over a finite field is a CM abelian variety. We will explain what the technical term CM, "complex multiplication",means precisely. Can it be CM lifted to characteristic zero? In joint work with Brian Conrad and Ching-Li Chai we give a complete answer to various ways this question can be phrased. In my talk explicit examples and general methods will be discussed, and proofs (some are easy, some quite involved) will be given.