Optical spectroscopy is a powerful approach for detecting defects and impurities in ZnO, an important electronic material. However, knowledge of how common optical signals are linked with defects and impurities is still limited. The Cu-related green luminescence is among the best understood luminescence signals, but theoretical descriptions of Cu-related optical processes have not agreed with experiment. Regarding native defects, assigning observed lines to specific defects has proven very difficult. Using first-principles calculations, we calculate the properties of native defects and impurities in ZnO and their associated optical signals. Oxygen vacancies are predicted to give luminescence peaks lower than 1 eV; while related zinc dangling bonds can lead to luminescence near 2.4 eV. Zinc vacancies lead to luminescence peaks below 2 eV, as do the related oxygen dangling bonds. However, when complexed with hydrogen impurities, zinc vacancies can cause higher-energy transitions, up to 2.3 eV. We also find that the Cu-related green luminescence is related to a (+/0) deep donor transition level.