A few years ago Perer Ozsvath and Zoltan Szabo introduced an invariant of 3-manifolds that they called "Heegaard Floer Homology". Taking the form of a bi-graded homology theory, this invariant has proved surprisingly adaptable. It can be used as a knot invariant which, among other things, carries the Alexander polynomial as its graded Euler characteristic. In a recent paper of Ozsvath and Szabo an invariant of contact structures was discovered in the Heegaard Floer homology of a contact 3-manifold $(M,\xi)$. This invariant is nonzero for Stein-fillable contact structures, and vanishes for overtwisted ones. In this collection of talks, I plan to discuss some of the basic definitions in Heegaard Floer homology and explain how one might use it to find new tight contact 3-manifolds.