Zinc oxide, a wide-band-gap semiconductor with many technological applications, typically exhibits n-type conductivity. The cause of this conductivity has been widely debated. A first-principles investigation, based on density functional theory, produces strong evidence that hydrogen acts as a source of conductivity: it can incorporate in high concentrations and behaves as a shallow donor. This behavior is unexpected and very different from hydrogen's role in other semiconductors, in which it acts only as a compensating center and always counteracts the prevailing conductivity. These insights have important consequences for control and utilization of hydrogen in oxides in general.