"Heegaard Floer homology" is an invariant for low-dimensional objects (knots in three-space, three-manifolds, and four-manifolds) defined using methods from symplectic geometry. For knots in three-space, this invariant has an elementary description which can be explicitly computed from a suitable combinatorial representation of the knot (called a "grid diagram"). I will explain this presentation of knot Floer homology, and then discuss some of its applications to problems in knot theory. I will describe here joint work with many coauthors, including: Robert Lipshitz, Ciprian Manolescu, Zoltan Szabo, and Dylan Thurston.