Could crystalline chiral condensates form in the core of compact stars? Can their existence be determined through astrophysical observations? In my talk, I will try to partially answer these questions. For this, I will first discuss the influence of some realistic astrophysical conditions on the mechanism of inhomogeneous chiral symmetry breaking and the structure of the energetically favored state of quark matter. I will then discuss possible observables related to the formation of crystalline condensates, with a particular emphasis on the effects on the equation of state of dense quark matter and its consequences on the resulting mass-radius sequences for compact stars.