We present first-principles electronic structure calculations of Mn doped III-V semiconductors based on the local spin-density approximation (LSDA) as well as the self-interaction corrected local spin density method (SIC-LSD). We find that it is crucial to use a self-interaction free approach to properly describe the electronic ground state. The SIC-LSD calculations predict the proper electronic ground state configuration for Mn in GaAs, GaP, and GaN. Excellent quantitative agreement with experiment is found for magnetic moment and p-d exchange in (GaMn)As. These results allow us to validate commonly used models for magnetic semiconductors. Furthermore, we discuss the delicate problem of extracting binding energies of localized levels from density functional theory calculations. We propose three approaches to take into account final state effects to estimate the binding energies of the Mn-d levels in GaAs. We find good agreement between computed values and estimates from photoemisison experiments.