The modern development of contact geometry in 3 dimensions has seen several (due to Giroux, Wendl, Latschev and Wendl, Hutchings, and others) invariants of contact structures meant in some sense to measure non-(Stein/symplectic)-fillability of the structure. We will discuss a new approach which uses Heegaard Floer homology to define an invariant with a similar aim, but which has several desirable properties lacking in earlier approaches. Time permitting, we will discuss some examples and applications. This is joint with joint work with Kutluhan, Matic, and Van Horn-Morris.