This paper deals with the asymptotic behavior of solutions to the delayed monostable equation: (∗) ut(t, x) = uxx(t, x)−u(t, x)+g(u(t− h, x)), x ∈ R, t> 0; here h > 0 and the reaction term g : R+ → R+ is Lipschitz continuous and has exactly two fixed points (zero and κ > 0). Under certain condition on the derivative of g at κ (without assuming classic KPP condition for g) the global stability of fast semiwavefronts is proved. Also, when the Lipschitz constant Lg is equal to g (0) the stability of all semi-wavefronts (e.g., critical, non-critical and asymptotically periodic semi-wavefronts) on each interval in the form (−∞, N], N ∈ R, to (∗) is established, which includes classic equations such as the Nicholson’s model.