  An increasing number of model results suggests that chiral symmetry is broken
inhomogeneously in a certain window at intermediate densities in the QCD phase
diagram. This could have significant effects on the properties of compact
stars, possibly leading to new astrophysical signatures. In this contribution
we discuss this idea by reviewing recent results on inhomogeneous chiral
symmetry breaking under an astrophysics-oriented perspective. After introducing
two commonly studied spatial modulations of the chiral condensate, the chiral
density wave and the real kink crystal, we focus on their properties and their
effect on the equation of state of quark matter. We also describe how these
crystalline phases are affected by different elements which are required for a
realistic description of a compact star, such as charge neutrality, the
presence of magnetic fields, vector interactions and the interplay with
color-superconductivity. Finally, we discuss possible signatures of
inhomogeneous chiral symmetry breaking in the core of compact stars,
considering the cases of mass-radius relations and neutrino emissivity
explicitly.
