In this paper, a multistage susceptible-infectious-recovered model with distributed delays and nonlinear incidence rate is investigated, which extends the model considered by Guo et al. [H. Guo, M. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J.Appl.Math., 72 (2012), 261-279]. Under some appropriate and realistic conditions, the global dynamics is completely determined by the basic reproduction number R-0. If R(0)1, then the infection-free equilibrium is globally asymptotically stable and the disease dies out in all stages. If R-0>1, then a unique endemic equilibrium exists, and it is globally asymptotically stable, and hence the disease persists in all stages. The results are proved by utilizing the theory of non-negative matrices, Lyapunov functionals, and the graph-theoretical approach. Copyright (c) 2016 John Wiley & Sons, Ltd.