The discovery of spin crossovers in iron under pressure in ferropericlase and iron-bearing magnesium silicate perovskite, the majormineral phases of the Earth's lower mantle, ignited in the last eight years intense discussions on the nature of this phenomenon and potential effects on the state and evolution of the Earth's mantle and of the planet as a whole. The nature of spin crossover in ferropericlase is better understood, and consensus on the interpretation of experimental data has been essentially achieved. However, in perovskite numerous experiments and computations have been carried out, often giving different results or suggesting different interpretations. Perovskite is a much more challenging system to investigate for a couple of reasons: there are two types of iron, ferrous (Fe2+) and ferric (Fe3+), more than one site can be occupied by iron, the A and B perovskite sites (as in ABX 3), and experiments do not probe the spin state unambiguously. Calculations are also especially difficult. This type of problem has challenged condensed matter theorists for decades. In addition, novel density functional theory + Hubbard U (DFT+U) electronic structure techniques, sometimes previously untested, are being used to address this type of problem. In summary, the challenge is multi-faceted. Here we review a series of calculations that have succeeded in reproducing experimental data and have been essential for their interpretation. They used the recently developed DFT+Usc method where the Hubbard U is obtained self-consistently by first principles. In particular, the electric field gradients (EFGs) at the nucleus of iron and thus quadrupole splittings (QSs) have been successfully computed, making direct comparisons between measured (via Mössbauer spectroscopy) and computed QSs for various spin states possible. This strategy led to unexpected findings in (Mg 1-xFex)SiO3 and in (Mg1-xFe x)(Si1-xFex)O3 (x = 0.125): in the pressure range of the lower mantle a) ferrous iron in the A site undergoes a site change, not a spin-state change and b) in ferric iron, spin crossover happens only in the B-site.