Considering the importance of time delay phenomenon in disease transmission and the interdependence of time delay and spatial location, a reaction-diffusion SIR epidemic model with nonlocal time delay (or time-space time delay) is proposed and the travelling wave solutions are discussed. Specifically, we define the basic reproduction number R 0 and the critical wave speed c ∗ . For every wave speed c ≥ c ∗ , the existence of travelling wave solutions is studied by using the upper–lower solutions, the fixed-point theorem and some limit techniques when R 0 > 1 . The nonexistence of travelling waves when R 0 > 1 for any 0 < c < c ∗ or R 0 < 1 for any c>0 is also proved. Finally, the influence of time-delay on disease propagation, especially the critical wave speed, is discussed through analysis and simulations.