Cold dense quark matter is a colour superconductor. At high densities the preferred phase is the colour-flavour-locked (CFL) phase with equal numbers of up, down and strange quarks. At intermediate densities (as in a neutron star), the nonzero strange quark mass upsets the SU(3) flavour symmetry of the CFL phase and unlocked colour superconducting phases can prevail. One unlocked phase is the crystalline phase, in which the Cooper pair condensate spontaneously breaks translational invariance by forming a face-centred-cubic pattern (this FCC pattern might also be detected in a crystalline superfluid of cold fermionic atoms). Another unlocked phase is the single-flavour phase, a J = 1 condensate that spontaneously breaks rotational invariance. I will describe these new phases and discuss implications for the physics of compact stars.