The QCD axion is probably the most robust solution to the strong CP problem and a natural dark matter candidate. While its properties, such as the mass and the couplings, are mostly determined by non-perturbative QCD effects I will present recent computations demonstrating that they can be reliably extracted with percent accuracy, which is important for the theory interpretation of experimental results. I will also discuss some recent lattice QCD results highlighting a departure from the usual instanton estimates and potentially changing the prediction for the axion relic abundance substantially. In this seminar I will discuss what changes to analysis strategy and detector setup will be necessary to definitively discover or exclude the experimentally-elusive pure-higgsino thermal relic at the FCC, thus closing the window on neutralino dark matter.