QCD at finite density and color superconductivity

The paper contains a brief review of recent applications of many-body theory to quark matter. We discuss the progress in theory of dense quark matter during the last two years, especially color superconductivity. We emphasize that there are two basic dynamical reasons for it: short-range forces induced by instantons and long-range ones mediated by exchanges of magnetic gluons. For quark matter which is supposed to be found in neutron stars, both lead to superconducting gaps on the order of 100 MeV. The most surprising facts are the rather impressive richness of different phases and their robustness in respect to variation of the fundamental interaction.