Recent years have seen many new invariants of manifolds, invariants defined using moduli spaces of solutions to certain partial differential equations. These include the anti-self-dual equations and Donaldson invariants, the pseudo-holomorphic curve equations of Gromov and the resulting Gromov and Gromov-Witten invariants, quantum cohomology, various Chern-Simon's invariants, Floer homology, and, the subject of this article, the Seiberg-Witten invariants. We cover the background necessary to define these invariants, give the definition, and then several applications to the topology of smooth four-manifolds including to symplectic four-manifolds and complex algebraic surfaces.