  We describe the effective lagrangian approach to the color superconductivity.
The effective description that arises if one considers only the leading terms
in the expansion for very high densities is particularly simple. It is based on
a lagrangian whose effective fermion fields are velocity-dependent; moreover
strong interactions do not change quark velocity and the effective lagrangian
does not contain spin matrices. All these features render the effective theory
similar to the Heavy Quark Effective Theory, which is the limit of Quantum
ChromoDynamics for infinite quark masses. For this reason one can refer to the
effective lagrangian at high density as the High Density Effective Theory
(HDET). In some cases HDET results in analytical, though approximate, relations
that are particularly simple to handle. After a pedagogical introduction,
several topics are considered. They include the treatment of the
Color-Flavor-Locking and the 2SC model, with evaluation of the gap parameters
by the Nambu-Gorkov equations, approximate dispersion laws for the gluons and
calculations of the Nambu-Goldstone Bosons properties. We also discuss the
effective lagrangian for the crystalline color superconductive (LOFF) phase and
we give a description of the phonon field related to the breaking of the
rotational and translational invariance. Finally a few astrophysical
applications of color superconductivity are discussed.
