Kesterite semiconductors, derived from the mineral Cu2(Zn,Fe)SnS4, adopt superstructures of the zincblende archetype. This family of semiconductors is chemically flexible with the possibility to tune the physical properties over a large range by modifying the chemical composition, while preserving the same structural backbone. In the simplest case, three metals (e.g. Cu, Zn and Sn) occupy the cation sublattice, which gives rise to a range of competing orderings (polymorphs) and the possibility for order–disorder transitions. The rich physics of the sulphide, selenide, and mixed-anion materials make them attractive for computer simulations in order to provide deeper insights and to direct experiments to the most promising material combinations and processing regimes. This topical review assesses the status of first-principles electronic structure calculations, optical modelling, and photovoltaic device simulations of kesterite semiconductors. Recent progress is discussed, and immediate challenges are outlined, in particular towards overcoming the voltage deficit in Cu2ZnSnS4 and Cu2ZnSnSe4 solar cells.