  QCD axions are at the crossroads of QCD topology and Dark Matter searches. We
present here the current status of topological studies on the lattice, and
their implication on axion physics. We outline the specific challenges posed by
lattice topology, the different proposals for handling them, the observable
effects of topology on the QCD spectrum and its interrelation with chiral and
axial symmetries. We review the transition to the Quark Gluon Plasma, the fate
of topology at the transition, and the approach to the high temperature limit.
We discuss the extrapolations needed to reach the regime of cosmological
relevance, and the resulting constraints on the QCD axion.
