Heegaard Floer homology is an invariant for three-manifolds (and four-manifolds) defined using Heegaard diagrams and methods from symplectic geometry, in joint work with Zoltan Szabo. I will give an overview of this construction, the primary properties of these invariants, and some of their topological applications. Time permitting, I will also describe some recent work with Robert Lipshitz and Dylan Thurston, where we define a version for three-manifolds with boundary.