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I review arguments for the existence of a critical point in the QCD phasediagram as a function of temperature and baryon chemical potential. I describehow heavy ion collision experiments at the SPS and RHIC can discover thetell-tale signatures of such a critical point, thus mapping this region of theQCD phase diagram. I then review the phenomena expected in cold dense quarkmatter: color superconductivity and color-flavor locking. I close with asnapshot of ongoing explorations of the implications of recent developments inour understanding of cold dense quark matter for the physics of compact stars.Comment: 29 pages. 4 figures. latex with aipproc. References added and  corrected. One version of this review is to appear in the Comments of Nuclear  and Particle Physics section of Comments on Modern Physics. Other versions  were contributed to the proceedings of the Conference on Intersections of  Nuclear and Particle Physics, Quebec, May 2000 and to the proceedings of the  40th Zakopane School of Theoretical Physics, June 200

The phases of QCD in heavy ion collisions and compact stars