In this paper, we mainly investigate the existence and nonexistence of traveling waves for a diffusive epidemic model with a general nonlinear incidence rate and infection-age structure. It is observed that whether the disease can spread or not depends on the basic reproduction number R 0 and critical wave speed c ∗ . More precisely, the traveling wave solution exists when R 0 > 1 and c ≥ c ∗ , while the traveling wave solution vanishes when R 0 ≤ 1 or R 0 > 1 and c ∈ ( 0 , c ∗ ) . By constructing a new pair of upper and lower solutions, an open problem proposed by Ducrot and Magal (Proc R Soc Edinb-A 139:459–482, 2009; Nonlinearity 24:2891–2911, 2011) is solved. It is also shown that infection-age structure can reduce the speed of disappearance of infectious diseases. We also investigate the effects of nonlinear incidence rate and age structure on the basic reproduction number and critical wave speed. Our results generalize some known results.