The properties of the QCD axion can be studied from the
dependence of strong interactions on the $\theta$-parameter. We present a study of the moments of the topological charge for
temperatures up to around 600 MeV, obtained by lattice QCD simulations with 2+1 flavors at the physical quark masses. Our numerical results for the topological susceptibility, when compared to the instanton gas calculation, differ both in size and in the temperature dependence. We discuss the implications of these results for axion phenomenology, a comparison with other more recent studies
and the prospects for future investigations.