I will seek to convince the astrophysicists in the audience that the phases of QCD matter. I'll describe properties of different phases, asking in each case how these properties would affect compact stars should the phase in question be present therein. By now, theorists have a good understanding of the "CFL" phase that occurs at asymptotic densities: it is a color superconductor, an ordinary superfluid, an electromagnetic insulator, and its response to neutrinos is dominated by its bosonic collective excitations. Compact stars are, however, not asymptotically dense. In the realistic regime, the theoretical situation is unsettled, with many phases still competing for theorists' attentions. These phases are all color superconductors but in other respects enumerated above they have properties in some way "opposite" to those of CFL matter.