Hybrid AMX 3 perovskites (A = Cs, CH 3NH 3; M = Sn, Pb; X = halide) have revolutionized the scenario of emerging photovoltaic technologies, with very recent results demonstrating 15% efficient solar cells. The CH 3NH 3PbI 3/MAPb(I 1−xCl x) 3 perovskites have dominated the field, while the similar CH 3NH 3SnI 3 has not been exploited for photovoltaic applications. Replacement of Pb by Sn would facilitate the large uptake of perovskite-based photovoltaics. Despite the extremely fast progress, the materials electronic properties which are key to the photovoltaic performance are relatively little understood. Density Functional Theory electronic structure methods have so far delivered an unbalanced description of Pb- and Sn-based perovskites. Here we develop an effective GW method incorporating spin-orbit coupling which allows us to accurately model the electronic, optical and transport properties of CH 3NH 3SnI 3 and CH 3NH 3PbI 3, opening the way to new materials design. The different CH 3NH 3SnI 3 and CH 3NH 3PbI 3 electronic properties are discussed in light of their exploitation for solar cells, and found to be dominantly due to relativistic effects. These effects stabilize the CH 3NH 3PbI 3 material towards oxidation, by inducing a deeper valence band edge. Relativistic effects, however, also increase the material band-gap compared to CH 3NH 3SnI 3, due to the valence band energy downshift (~0.7 eV) being only partly compensated by the conduction band downshift (~0.2 eV).