Given a group G, a set of variables x1, ..., xn, and a word in the xi and their inverses, one can define the evaluation map or *word map* from Gn to G. There has been a flurry of recent interest in results asserting that under various hypotheses, word maps are surjective or at least have large image. I will discuss some of these results, the varied techniques (from algebraic geometry, analytic number theory, and harmonic analysis) which have been employed, and a number of questions which remain open.