Motivated by the uniqueness problem for monostable semi-wave -fronts, we propose a revised version of the Diekmann and Kaper theory of a nonlinear convolution equation. Our version of the Diekmann–Kaper theory allows (1) to consider new types of models which include nonlocal KPP type equations (with either symmetric or anisotropic dispersal), nonlocal lattice equations and delayed reaction–diffusion equations; (2) to incorporate the critical case (which corresponds to the slowest wavefronts) into the consideration; (3) to weaken or to remove various restrictions on kernels and nonlinearities. The results are compared with those of Schumacher (J Reine Angew Math 316: 54–70, 1980), Carr and Chmaj (Proc Am Math Soc 132: 2433–2439, 2004), and other more recent studies.