Abstract: Heegaard Floer homology is an invariant for three-manifolds, and four-manifolds, which is defined by counting pseudo-holomorphic curves in a symplectic manifold associated to a Heegaard diagram. I will describe these invariants, along with some more recently-discovered purely combinatorial formulations (in certain special cases), and sketch some topological applications of the theory. Heegaard Floer homology was defined in joint work with Zoltan Szabo; some of the combinatorial formulations include joint work with Zoltan and also Ciprian Manolescu, Sucharit Sarkar, and Dylan Thurston.