  We prove the existence of a continuous family of positive and generally
non-monotone travelling fronts in delayed reaction-diffusion equations
$u_t(t,x) = \Delta u(t,x)- u(t,x) + g(u(t-h,x)) (*)$, when $g \in C^2(R_+,R_+)$
has exactly two fixed points: $x_1= 0$ and $x_2= a >0$. Recently, non-monotonic
waves were observed in numerical simulations by various authors. Here, for a
wide range of parameters, we explain why such waves appear naturally as the
delay $h$ grows. For the case of $g$ with negative Schwarzian, our conditions
are rather optimal; we observe that the well known Mackey-Glass type equations
with diffusion fall within this subclass of $(*)$. As an example, we consider
the diffusive Nicholson's blowflies equation.
