Summary: We study the wavefront solutions of the scalar reaction-diffusion equations \[ u_{t}(t,x)=\Delta u(t,x)-u(t,x)+g(u(t-h,x)) \] with monotone reaction term \(g: \mathbb{R}_{+} \mathbb{R}_+\) 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 \(h>0\). To this end, we describe the asymptotics of all wavefronts (including critical and non-critical fronts) at \(-\infty\). We also prove the uniqueness of wavefronts (up to a translation). In addition, a new uniqueness result for a class of nonlocal lattice equations is presented.