  We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e.
PDEs with $n$ independent variables) with one unknown function, viewed as
hypersurfaces $\mathcal{E}$ in the Lagrangian Grassmann bundle $M^{(1)}$ over a
$(2n+1)$-dimensional contact manifold $(M,\mathcal{C})$. We develop the theory
of characteristics of the equation $\mathcal{E}$ in terms of contact geometry
and of the geometry of Lagrangian Grassmannian and study their relationship
with intermediate integrals of $\mathcal{E}$. After specifying the results to
general Monge-Amp\`ere equations (MAEs), we focus our attention to MAEs of type
introduced by Goursat, i.e. MAEs of the form $$ \det|\frac{\partial^2
f}{\partial x^i\partial x^j}-b_{ij}(x,f,\nabla f)\|=0. $$ We show that any MAE
of the aforementioned class is associated with an $n$-dimensional
subdistribution $\mathcal{D}$ of the contact distribution $\mathcal{C}$, and
viceversa. We characterize this Goursat-type equations together with its
intermediate integrals in terms of their characteristics and give a criterion
of local contact equivalence. Finally, we develop a method of solutions of a
Cauchy problem, provided the existence of a suitable number of intermediate
integrals.
