  Let $X$ be a hypersurface, of degree $d$, in an $n$--dimensional projective
space. The Hessian map is a rational map from $X$ to the projective space of
symmetric matrices that sends a point $p\in X$ to the Hessian matrix of the
defining polynomial of $X$ evaluated at $p$. The Hessian correspondence is the
map that sends a hypersurface to its Hessian variety; i.e. the Zariski closure
of its image via the Hessian map. In this paper, we study this correspondence
for hypersurfaces with Waring rank at most $n+1$ and for hypersurfaces of
degree $3$ and $4$. We prove that, for hypersurfaces with Waring rank $k\leq
n+1$, the map is birational onto its image for $d$ even, and it is generically
finite of degree $2^{k-1}$ for $d$ odd. We prove that, for degree $3$ and
$n=1$, the map is two to one, and that, for degree $3$ and $n\geq 2$, and for
degree $4$, the Hessian correspondence is birational. In this study, we
introduce the $k$--gradients varieties and analyze their main properties. We
provide effective algorithms for recovering a hypersurface from its Hessian
variety, for degree $3$ and $n\geq 1$, and for degree $4$ and $n$ even.
