In this Letter, a generalization of pairwise models to non-Markovian epidemics on networks is presented. For the case of infectious periods of fixed length, the resulting pairwise model is a system of delay differential equations, which shows excellent agreement with results based on stochastic simulations. Furthermore, we analytically compute a new R_{0}-like threshold quantity and an analytical relation between this and the final epidemic size. Additionally, we show that the pairwise model and the analytic results can be generalized to an arbitrary distribution of the infectious times, using integro-differential equations, and this leads to a general expression for the final epidemic size. By showing the rigorous link between non-Markovian dynamics and pairwise delay differential equations, we provide the framework for a more systematic understanding of non-Markovian dynamics.