  This paper concerns the semi-wavefronts (i.e. bounded solutions $u=\phi(x \nu
+ct) >0,$ $ |\nu|=1, $ satisfying $\phi(-\infty)=0$) to the delayed KPP-Fisher
equation $$u_t(t,x) = \Delta u(t,x) + u(t,x)(1-u(t-\tau,x)), \ u \geq 0,\ x \in
\R^m. \eqno(*)$$ First, we show that each semi-wavefront should be either
monotone or slowly oscillating. Then a complete solution to the problem of
existence of semi-wavefronts is provided. We prove next that the
semi-wavefronts are in fact wavefronts (i.e. additionally $\phi(+\infty)=1$) if
$c \geq 2$ and $\tau \leq 1$; our proof uses dynamical properties of some
auxiliary one-dimensional map with the negative Schwarzian. The analysis of the
fronts' asymptotic expansions at infinity is another key ingredient of our
approach. It allows to indicate the maximal domain ${\mathcal D}_n$ of
$(\tau,c)$ where the existence of non-monotone wavefronts can be expected. Here
we show that the problem of wavefront's existence is closely related to the
Wright's global stability conjecture.
