Hybrid organometal halide perovskites have been demonstrated to have outstanding performance as semiconductors for solar energy conversion. Further improvement of the efficiency and stability of these devices requires a deeper understanding of their intrinsic photophysical properties. Here, the structural and optical properties of high-quality single crystals of CH3NH3PbI3 from room temperature to 5 K are investigated. X-ray diffraction reveals an extremely sharp transition at 163 K from a twinned tetragonal I4/mcm phase to a low-temperature phase characterized by complex twinning and possible frozen disorder. Above the transition temperature, the photoluminescence is in agreement with a band-edge transition, explaining the outstanding performances of the solar cells. Whereas below the transition temperature, three different excitonic features arise, one of which is attributed to a free-exciton and the other two to bound excitons (BEs). The BEs are characterized by a decay dynamics of about 5 mu s and by a saturation phenomenon at high power excitation. The long lifetime and the saturation effect make us attribute these low temperature features to bound triplet excitons. This results in a description of the room temperature recombination as being due to spontaneous band-to-band radiative transitions, whereas a diffusion-limited behavior is expected for the low-temperature range.