Zinc oxide (ZnO) is a wide band gap semiconductor with potential applications in optoelectronics, transparent electronics, and spintronics. The high efficiency of UV emission in this material could be harnessed in solid-state white lighting devices. The problem of defects, in particular, acceptor dopants, remains a key challenge. In this review, defects in ZnO are discussed, with an emphasis on the physical properties of point defects in bulk crystals. As grown, ZnO is usually n-type, a property that was historically ascribed to native defects. However, experiments and theory have shown that O vacancies are deep donors, while Zn interstitials are too mobile to be stable at room temperature. Group-III (B, Al, Ga, and In) and H impurities account for most of the n-type conductivity in ZnO samples. Interstitial H donors have been observed with IR spectroscopy, while substitutional H donors have been predicted from first-principles calculations but not observed directly. Despite numerous reports, reliable p-type conductivity has not been achieved. Ferromagnetism is complicated by the presence of secondary phases, grain boundaries, and native defects. The famous green luminescence has several possible origins, including Cu impurities and Zn vacancies. The properties of group-I (Cu, Li, and Na) and group-V (N, P, As, and Sb) acceptors, and their complexes with H, are discussed. In the future, doping of ZnO nanocrystals will rely on an understanding of these fundamental properties.