Given an abelian variety over a p-adic field, when is its reduction mod p again an abelian variety? I will make this question precise and then give a solution. Along the way, I will digress to talk about Neron models (without proofs). If time permits, I will discuss some further questions along these lines and some applications. The talk should be relatively accessible; the phrase "elliptic curve" can be freely substituted for "abelian variety".