The complex of curves on a surface S is a simplicial complex that encodes the structure of the set of homotopy classes of simple curves on S. This complex, viewed as a metric space, turns out to be hyperbolic in the sense of Gromov, and this has applications to the geometry of hyperbolic 3-manifolds, the structure of the mapping class groups, and (more tentatively) to the study of Heegaard splittings.