Moduli spaces and parameter spaces for geometric objects such as curves, bundles, etc., are two related central topics in algebraic geometry. They can be as simple as projective spaces and Grassmannians; they can also be as complicated as the moduli spaces of stable maps. The former are smooth; the latter can contain arbitrary singularities. In this talk, using examples, I will introduce various moduli spaces and parameter spaces, and explain their connections with some central problems in algebraic geometry. Toward the end, I will explain my joint program with Jun Li (Stanford) to modularly desingularize the moduli spaces of stable maps; the results may be applied to the central problem of resolution of singularities as well as to the difficult high genus Gromov-Witten theory.