Nanostructured semiconductors emit light from electronic states known as excitons(1). For organic materials, Hund's rules(2) state that the lowest-energy exciton is a poorly emitting triplet state. For inorganic semiconductors, similar rules(3) predict an analogue of this triplet state known as the 'dark exciton'(4). Because dark excitons release photons slowly, hindering emission from inorganic nanostructures, materials that disobey these rules have been sought. However, despite considerable experimental and theoretical efforts, no inorganic semiconductors have been identified in which the lowest exciton is bright. Here we show that the lowest exciton in caesium lead halide perovskites (CsPbX3, with X = Cl, Br or I) involves a highly emissive triplet state. We first use an effective-mass model and group theory to demonstrate the possibility of such a state existing, which can occur when the strong spin-orbit coupling in the conduction band of a perovskite is combined with the Rashba effect(5-10). We then apply our model to CsPbX3 nanocrystals(11), and measure size-and composition-dependent fluorescence at the single-nanocrystal level. The bright triplet character of the lowest exciton explains the anomalous photon-emission rates of these materials, which emit about 20 and 1,000 times faster(12) than any other semiconductor nanocrystal at room(13-16) and cryogenic(4) temperatures, respectively. The existence of this bright triplet exciton is further confirmed by analysis of the fine structure in low-temperature fluorescence spectra. For semiconductor nanocrystals, which are already used in lighting(17), lasers(18) and displays(19), these excitons could lead to materials with brighter emission. More generally, our results provide criteria for identifying other semiconductors that exhibit bright excitons, with potential implications for optoelectronic devices.