Halide perovskites constitute a chemically diverse class of crystals with great promise as photovoltaic absorber materials, featuring band gaps between about 1 and 3.5 eV depending on composition. Their diversity calls for a general computational approach to predicting their band gaps. However, such an approach is still lacking. Here, we use density functional theory (DFT) and ab initio many-body perturbation theory within the GW approximation to compute the quasiparticle or fundamental band gap of a set of ten representative halide perovskites: CH3NH3PbI3 (MAPbI3), MAPbBr3, CsSnBr3, (MA)2BiTlBr6, Cs2TlAgBr6, Cs2TlAgCl6, Cs2BiAgBr6, Cs2InAgCl6, Cs2SnBr6, and Cs2Au2I6. Comparing with recent measurements, we find that a standard generalized gradient exchange-correlation functional can significantly underestimate the experimental band gaps of these perovskites, particularly in cases with strong spin-orbit coupling (SOC) and highly dispersive band edges, to a degree that varies with composition. We show that these nonsystematic errors are inherited by one-shot G0W0 and eigenvalue self-consistent GW0 calculations, demonstrating that semilocal DFT starting points are insufficient for MAPbI3, MAPbBr3, CsSnBr3, (MA)2BiTlBr6, Cs2TlAgBr6, and Cs2TlAgCl6. On the other hand, we find that DFT with hybrid functionals leads to an improved starting point and GW0 results in better agreement with experiment for these perovskites. Our results suggest that GW0 with hybrid functional-based starting points are promising for predicting band gaps of systems with large SOC and dispersive bands in this technologically important class of semiconducting crystals.