I will describe an extension of the U=0 specialization of Heegaard Floer homology, HF, to three-manifolds with parameterized boundary. The resulting theory, bordered Floer homology, associates a differential algebra to an oriented two-manifold, and a differential module module over that algebra to a three-manifold with boundary. These differential modules can also be used to reconstruct HF for closed three-manifolds. This is joint work with Robert Lipshitz and Dylan Thurston.