A Floer homology theory is, roughly speaking, a homology theory derived by running the machinery of Morse homology on a high- or infinite-dimensional space associated to a manifold in some way. These theories do not generally produce homology groups isomorphic to singular homology, but instead give information about structures on the manifold in question depending on the space associated. In this talk, we'll discuss the original formulation given by Andreas Floer in 1988, now known as Lagrangian Floer homology and used in Floer's proof of the Arnold conjecture in symplectic geometry, as well as the newer theory of Heegaard-Floer homology and its applications to 3- and 4-manifold topology.