ZnO is one of the top candidates for blue light optoelectronics because of its wide bandgap properties. However, fabricating high quality p-type ZnO has proven to be difficult. While none of the group-I doping yields p-type behavior and Nitrogen doping shows only limited success, doping with larger group-V elements, which should cause high strain and has low solubility on the oxygen site, show some preliminary surprising success. Based on first principles calculations, we have developed a model involving complex doping that could explain the observed p-type behavior in P-, As-, and Sb-doped ZnO and fits well with experimental growth/annealing conditions. If time permits, I will also present our recent study of “substitutional diatomic molecules” in ZnO. Small anions, such as O, N and C, prefer to form diatomic molecules on the Oxygen-site and behave as donors. Based on energetic and growth conditions, (N2)_O and (NC)_O can exist in N-doped ZnO in a large amount and could be the cause for the limited success of obtaining p-type doping using N-doped ZnO. The vibrational properties as well as electronic properties of these substitutional diatomic molecules resemble closely those of the free molecules.