Bismuth ferrite has been extensively researched in the last 15 years, in part because it is one of the very few ferroelectrics that also shows magnetic ordering at room temperature. It crystallizes in a rhombohedral perovskite structure where its cations are displaced from high-symmetry postions to give rise to one of the largest electric polarizations known in crystals, and where its oxygen octahedra are rotated with respect to those in the aristotype phase. Substituting iron by other transition metal also gives rise to perovskite-like structures, but the distortions in those are very different. For example, bismuth cobaltite crystallizes in a supertetragonal phase, while bismuth scandiate, bismuth chromate, and bismuth manganite crystallize in a non-polar structure where the bismuth atoms follow an antiferroelectric pattern. In this talk I will describe our findings that all those structures correspond in reality to a few of the energy minima that these materials show, and that it is only a delicate balance between the total energy of those structures what results in the rich behaviour we observe in this family of materials.