The Phases of QCD in Heavy Ion Collisions and Compact Stars

I review arguments for the existence of a critical point in the QCD phase diagram and describe how heavy ion collision experiments can discover the tell-tale signatures of such a critical point, thus mapping this region of the QCD phase diagram. I also review the phenomena expected in cold, dense quark matter. In this regime, quarks form Cooper pairs and QCD possesses a superconducting phase which may also break chiral symmetry. I close with a look at ongoing explorations of the implications of these recent developments in our understanding of cold dense quark matter for the physics of compact stars.