GaN p-i-n photodetectors grown on sapphire by reactive molecular beam epitaxy have been characterized by measurements of room-temperature current-voltage (I-V), temperature-dependent capacitance (C-V-T), and deep level transient spectroscopy (DLTS) under both majority and minority carrier injection. Due to what we believe to be threading dislocations, the reverse I-V curves of p-i-n photodetectors show typical electric-field enhanced soft breakdown characteristics. A carrier freeze-out due to the de-ionization of Mg-related deep accepters has been found by C-V-T measurements. Three electron traps, B (0.61 eV), D (0.23 eV), and E-1 (0.25 eV) and one hole trap, H-3 (0.79 eV) have been revealed by DLTS measurements. The photodetectors with lower leakage currents usually show higher responsivity and lower trap densities of D and E-1.