  At ultra-high density, matter is expected to form a degenerate Fermi gas of
quarks in which there is a condensate of Cooper pairs of quarks near the Fermi
surface: color superconductivity. In this chapter we review some of the
underlying physics, and discuss outstanding questions about the phase structure
of ultra-dense quark matter. We then focus on describing recent results on the
crystalline color superconducting phase that may be the preferred form of cold,
dense but not asymptotically dense, three-flavor quark matter. The gap
parameter and free energy for this phase have recently been evaluated within a
Ginzburg-Landau approximation for many candidate crystal structures. We
describe the two that are most favorable. The robustness of these phases
results in their being favored over wide ranges of density. However, it also
implies that the Ginzburg-Landau approximation is not quantitatively reliable.
We describe qualitative insights into what makes a crystal structure favorable
which can be used to winnow the possibilities. We close with a look ahead at
the calculations that remain to be done in order to make quantitative contact
with observations of compact stars.
