QCD at and near finite isospin density

The subject of this talk is the phase structure and the equation of state of strongly interacting matter in the presence of an isospin-asymmetry. If sufficiently large, this asymmetry between the up and down quark densities induces various new phases like a Bose-Einstein condensate of charged pions or a BCS-type superconductor made of up and anti-down quark pairs. The pion condensed phase can be particularly relevant for cosmology: recent studies indicate that it might have emerged during the evolution of the early Universe under certain circumstances compatible with observations. I will summarize the recent results for the phase diagram and the equation of state obtained by first-principles lattice QCD simulations, together with the conceptual and technical challenges that these entail. The potential astrophysical implications will also be discussed in some detail.