In this thesis, we first present a brief overview of various theoretical approaches used to examine the electronic structure of defects in GaN. Using the recently developed hybrid density functional theory (HSE06) along with the experimental measurements, we propose a new explanation of the nature of the yellow luminescence band in carbon-doped GaN. We conduct a systematic study of electronic and optical properties of defects (Carbon, Oxygen, Silicon related) that are candidates for the origin of yellow luminescence. We show that the CN-ON complex is significantly more likely to form compared to isolated carbon configurations. In contrast to the properties of the isolated carbon acceptor, calculated defect levels and optical transitions involving deep level of the CN-ON complex agree quite well with our thermal luminescence quenching data as well as with the experimentally measured C-doped GaN luminescence spectra. Hence, the CN-ON complex, rather than isolated C impurity, is more likely to resolve a long-standing problem of the yellow luminescence in GaN.