An emission band at 3.31eV is frequently observed in low-temperature photoluminescence (PL) measurements on ZnO p-doped with group-V elements, and also on nominally undoped ZnO layers and nanostructures. It has alternatively been ascribed to LO- or TO-phonon replicas of free excitons, to acceptor-bound excitons, to donor–acceptor pair transitions, to two-electron satellites of donor-bound excitons, or to free-to-bound transitions. This band frequently dominates the PL of ZnO nanostructures and layers at room temperature. Annealing leads to drastic changes in its intensity.We report on low-temperature cathodoluminescence measurements with very high spatial resolution and high-resolution transmission electron microscope investigations carried out on the same pieces of hetero-epitaxial ZnO samples with unusual layer orientation. These data allow us to correlate this emission unambiguously with c-plane stacking faults. The emission is found to be due to the recombination of a free electron with a hole bound to a relatively shallow acceptor state ≈130meV above the valence band edge. Locally, these acceptor states occur in high concentrations of up to some 1018cm−3, and thus lead to strong two-dimensional perturbations of the free carrier concentration. They have severe implications for the conductivity of layers and nanostructures in general, and on the interpretation of Hall and luminescence data in particular. Literature data are critically reviewed in the light of these findings.