Model criticism is a growing focus of research in stochastic epidemic modelling, following the successful addressing of model fitting and parameter estimation via powerful computationally intensive statistical methods in recent decades. In this paper, we consider a variety of stochastic representations of epidemic outbreaks, with emphasis on individual-based continuous-time models, and review the range of model comparison and assessment approaches currently applied. We highlight some of the factors that can serve to impede checking and criticism of epidemic models such as lack of replication, partial observation of processes, lack of prior knowledge on parameters in competing models, the nonnested nature of models to be compared, and computational challenges. Based on a wide selection of approaches as reported in the literature, we argue that assessment and comparison of stochastic epidemic models is complex and often, by necessity, idiosyncratic to specific applications. We particularly advocate following the advice of Box [J. Amer. Statist. Assoc. 71 (1976) 791–799] to be selective regarding the model inadequacies for which one tests and, moreover, to be open to the blending of classical and Bayesian ideas in epidemic model criticism, rather than adhering to a single philosophy.