Gromov-Witten theory probes the geometry of projective manifolds by studying maps into them, from algebraic curves. The theory is a natural source of cohomology classes on the moduli space of stable curves, and includes several classical constructions in algebraic geometry. I will explain a speculation, going back to Levine and Pandharipande, that the cohomology classes produced by Gromov-Witten theory always lie in a small and predictable part of cohomology, known as the tautological ring. I will then outline some recent advances that give us some control over the classes produced by Gromov-Witten theory, and then speculate on what this might mean for the speculation of Levine-Pandharipande.