This is a joint Symplectic Geometry - Topology seminar.   We define an invariant of contact structures in dimension three based on the contact invariant of Ozsvath and Szabo from Heegaard Floer homology. This invariant takes values in $\Z_{\geq0}\cup\{\infty\}$, is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, and non-decreasing under Legendrian surgery. This is joint work with Cagaty Kutluhan, Jeremy Van Horn-Morris and Andy Wand.