Summary: This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed \(c^\ast\) and the basic reproduction number \(R_0\) are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder’s fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when \(\mathcal{R}_0>1\), for every speed \(c>c^\ast\) there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any \(0<c< c^*\) when \(\mathcal{R}_0>1\) or \(c>0\) when \(\mathcal{R}_0<1\). Finally, we analyze the effects of nonlocal time delay on the minimum wave speed.