Gallium nitride (GaN) based alloys have evolved into the material class of choice for blue and white light-emitting diodes (LEDs). The reason is the large, tunable band gap, which allows light emission of nearly the whole visible spectrum. Improving the p-conductivity of GaN would allow for more efficient and brighter LEDs based on this material class. In the present PhD Thesis limitations in the p-doping of GaN were investigated. The examination is carried out by calculating defect energetics by means of density-functional theory (DFT), which is the state-of-the-art ab initio method for modeling and describing point defects. Large defects, such as the experimentally observed inversion domains (IDs) in p-type GaN:Mg, are not feasible to model within the framework of DFT. An alternative are coarse-grained methods employing accurate, atom-centered atomic orbitals. However, the construction of accurate and, in particular, transferable atomic orbital basis sets is far away from being trivial. Within this PhD thesis the QUAMOL concept is introduced, which constructs atom-centered, numerical orbitals based on plane-wave DFT calculations. The applicability and performance of the developed approach is demonstrated for semiconducting and metallic test systems, which show that the constructed orbitals are accurate and transferable. Further, the dominant point defects in GaN are studied in detail. Based on calculated formation energies nitrogen vacancies have been identified as possible compensators aside hydrogen in GaN:Mg. A tightrope walk between providing as much nitrogen as needed to avoid vacancy formation and providing as less nitrogen as possible to hinder the phase separation GaN/Mg3N2 at high Mg concentrations has finally been identified as the theoretical limitation of p-doping in GaN.