  We study the existence and uniqueness of wavefronts to the scalar
reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) +
g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h
>0$. We are mostly interested in the situation when the graph of $g$ is not
dominated by its tangent line at zero, i.e. when the condition $g(x) \leq
g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a
special type of rapidly decreasing wavefronts (pushed fronts) can appear in
non-delayed equations (i.e. with $h=0$). One of our main goals here is to
establish a similar result for $h>0$. We prove the existence of the minimal
speed of propagation, the uniqueness of wavefronts (up to a translation) and
describe their asymptotics at $-\infty$. We also present a new uniqueness
result for a class of nonlocal lattice equations.
