<p>This study is concerned with the existence of traveling-wave solutions to a temporally discrete predator–prey model with time delays and spatial diffusion. To overcome the difficulties caused by nonmonotone and discrete features, a new type of partial quasimonotone condition is introduced. By using the cross-iteration method and Schauder’s fixed-point theorem, the existence of traveling-wave solutions is reduced to the existence of a pair of upper and lower solutions to the corresponding profile equation. The obtained results can be well applied to temporally discrete reaction–diffusion Lotka–Volterra systems with delays.</p>