  In this paper we study the Hessian map $h_{d,r}$ which associates to any
hypersurface of degree $d$ in ${\mathbb P}^r$ its Hessian hypersurface. We
study general properties of this map and we prove that: $h_{d,1}$ is birational
onto its image if $d\geq 5$; we study in detail the maps $h_{3,1}$, $h_{4,1}$
and $h_{3,2}$; we study the restriction of the Hessian map to the locus of
hypersurfaces of degree $d$ with Waring rank $r+2$ in ${\mathbb P}^r$, proving
that this restriction is injective as soon as $r\geq 2$ and $d\geq 3$, which
implies that $h_{3,3}$ is birational onto its image; we prove that the
differential of the Hessian map is of maximal rank on the generic hypersurfaces
of degree $d$ with Waring rank $r+2$ in ${\mathbb P}^r$, as soon as $r\geq 2$
and $d\geq 3$.
