Crystalline color superconductors: A review

Non-homogenous superconductors and non-homogenous superfluids appear in a variety of contexts which include quark matter at extreme densities, fermionic systems of cold atoms, type-II cuprates and organic superconductors. In the present review we shall focus on the properties of quark matter at high baryonic density which can exist in the interior of compact stars. The conditions that are realized in this stellar objects tend to disfavor standard symmetric BCS pairing and may be in favor of a non-homogenous color superconducting phase. We discuss in details the properties of non-homogenous color superconductors and in particular of crystalline color superconductors. We also review the possible astrophysical signatures associated with the presence of non-homogenous color superconducting phases within the core of compact stars.