The performance of solar cells based on hybrid halide perovskites has seen an unparalleled rate of progress, while our understanding of the underlying physical chemistry of these materials trails behind. Superficially, CH3NH3PbI3 is similar to other thin-film photovoltaic materials: a semiconductor with an optical band gap in the optimal region of the electromagnetic spectrum. Microscopically, the material is more unconventional. Progress in our understanding of the local and long-range chemical bonding of hybrid perovskites is discussed here, drawing from a series of computational studies involving electronic structure, molecular dynamics, and Monte Carlo simulation techniques. The orientational freedom of the dipolar methylammonium ion gives rise to temperature-dependent dielectric screening and the possibility for the formation of polar (ferroelectric) domains. The ability to independently substitute on the A, B, and X lattice sites provides the means to tune the optoelectronic properties. Finally, ten critical challenges and opportunities for physical chemists are highlighted.