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We have theoretically investigated the origin of ultraviolet photoluminescence (PL) in ZnO quantum dots with diameters from 2 to 6 nm. Two possible sources of ultraviolet PL have been considered: excitons confined in the quantum dot and excitons bound to an ionized impurity located at the quantum-dot surface. It is found that depending on the fabrication method and surface passivation technique, the ultraviolet PL of ZnO quantum dots can be attributed to either confined excitons or surface-bound ionized acceptor-exciton complexes. The exciton radiative lifetime is shown to be very sensitive to the exciton localization and can be used as a tool to discriminate between these two sources of PL.