Exploring phases of dense QCD with compact stars

I review a number of recent developments in the physics of compact stars containing deconfined quark matter, including (a)~their cooling with possible phase transition from a fully gapped to a gapless phase of QCD at low temperatures and large isospin, (b)~the transport coefficients of the 2SC phase and the role played by the Aharonov-Bohm interactions between flux-tubes and unpaired fermions; ©~rapidly rotating compact stars and spin-down and spin-up induced phase transition between hadronic and QCD matter as well as between different phases of QCD.