Fibre-reinforced composites are rapidly increasing their market share in structural applications. Nevertheless, this increase would be much stronger if reliable failure predictions were available. These predictions are not only insufficiently reliable for complex loading of multidirectional composites, but even for longitudinal tensile failure of unidirectional (UD) composites. Since composite failure usually coincides with longitudinal failure of a 0° ply, the reliability often hinges on longitudinal failure predictions of UD composites. Despite great progress in the state-of-the-art models, significant obstacles remain in collecting the necessary input data and understanding the influence of the modelling assumptions. This review therefore surveys the mechanics, chemistry and physics involved in tensile failure of UD composites and highlights potential areas for improvement. Specific proposals are made to advance the state-of-the-art strength models, which could catalyse the use of composites in structural applications.