Materials with the perovskite ABX3 structure play a major role across materials chemistry and physics as a consequence of their ubiquity and wide range of useful properties. ReO3-type structures can be described as ABX3 perovskites in which the A-cation site is unoccupied, giving rise to the general composition BX3, where B is typically a cation and X is a bridging anion. The chemical diversity of such structures is extensive, ranging from simple oxides and fluorides, such as WO3 and AlF3, to complex structures in which the bridging anion is polyatomic, such as in the Prussian blue-related cyanides Fe(CN)3 and CoPt(CN)6. The same ReO3-type structure is found in metal-organic frameworks, for example, ln (im)3(im = imidazolate) and the well-known MOF-5 structure, where the B-site cation is polyatomic. The extended 3D connectivity and openness of this structure type leads to compounds with interesting and often unusual properties. Notable among these properties are negative thermal expansion (for example, ScF3), photocatalysis (for example, CoSn(OH)6), thermoelectricity (for example, CoAs3) and superconductivity in a phase that is controversially described as SH3 with a doubly interpenetrating ReO3 structure. We present an account of this exciting family of materials and discuss future opportunities in the area.