The famous Witten-Veneziano formula for the mass of the eta' meson links the fluctuations of the topological charge density in QCD to an experimentally accessible quantity. As it stands, the formula is however affected by short-distance singularities, whose subtraction leads to a non-perturbative ambiguity on the QCD side of the equation. The issue proved to be a persistent one, which could only be resolved after a series of important theoretical advances in lattice and continuum QCD were made. The most relevant of these are reviewed, focussing on the concepts rather than on any technical details.