Let (X0,2 o) be a polarized abelian variety over a field of characteristic p, p + 0; let R be a domain of characteristic zero. We say (X 0, 20) lifts to R provided there is a polarized abelian scheme, (X, 2), over R and a k-valued point of Spec R such that the fiber of(X, 2) over this point is (Xo, 20). Mumford ([5], [7]) has shown that any polarized abelian variety lifts to characteristic zero, i.e., for any polarized abelian variety we can find a characteristic zero domain R so that (Xo, 20) lifts to R. Mumford 's approach does not give information about the domain R. For example: Does an arbitrary polarized abelian variety lift to a characteristic zero domain that is unramified over p? The answer to this is no; Ogus found a polarization on the product of a supersingular elliptic curve with itself that fails to lift to a domain R ifp is not ramified in R. We present his example in w On the other hand using classical facts about lifting endomorphisms of elliptic curves, it is easy to show that Ogus' polarized abelian surface lifts if you allow x/P in R; furthermore the product of two supersingular elliptic curves allows a principal polarization and with this polarization we can accomplish lifting without ramification. This raises the questions: (A) How much ramification is needed to lift an arbitrary polarized abelian variety? (B) Which polarized abelian varieties lift without ramification? (C) Forgetting the polarization, can you lift an abelian variety to an abelian scheme over a characteristic zero domain in which p does not ramify? We answer (A) and (B) in this paper; (C) is, I think, still open. The answer to (A) is given in the