Zinc oxide (ZnO) is a semiconductor that emits bright UV light, with little wasted heat. This intrinsic feature makes it a promising material for energy-efficient white lighting, nano-lasers, and other optical applications. For devices to be competitive, however, it is necessary to develop reliable p-type doping. Although substitutional nitrogen has been considered as a potential p-type dopant for ZnO, recent theoretical and experimental work suggests that nitrogen is a deep acceptor and will not lead to p-type conductivity. In nitrogen-doped samples, a red photoluminescence (PL) band is correlated with the presence of deep nitrogen acceptors. PL excitation (PLE) measurements show an absorption threshold of 2.26 eV, in good agreement with theory. The results of these studies seem to rule out group-V elements as shallow acceptors in ZnO, contradicting numerous reports in the literature. Optical studies on ZnO nanocrystals show some intriguing leads. At liquid-helium temperatures, a series of sharp IR absorption peaks arise from an unknown acceptor impurity. The data are consistent with a hydrogenic acceptor 0.46 eV above the valence band edge. While this binding energy is still too deep for many practical applications, it represents a significant improvement over the 1.4-1.5 eV binding energy for nitrogen acceptors. Nanocrystals present another twist. Due to their high surface-to-volume ratio, surface states are especially important. In our model, the 0.46 eV level is shallow with respect to the surface valence band, raising the possibility of surface hole conduction.