The reductions modulo primes of a fixed elliptic curve defined over the rational numbers provide an interesting and useful generalization of the finite field Z/pZ. The study of Z/pZ as p varies has been a fertile source of problems in classical analytic number theory. Similar problems about elliptic curves are providing new challenges for modern analytic number theory. In this talk I will first review some of the analogies and especially the insights of Serre concerning these reductions. In order to understand them it is helpful to identify the Frobenius in explicit terms. This has some nice side applications for "non-abelian reciprocity" . It also leads to a set of problems where the classical sieve techniques break down and new ones must be found. These problems are often so difficult that the generalized Riemann hypothesis must be assumed in order to prove realistic results.