We will introduce the rudiments of a new theory of non-smooth
solutions which applies to fully nonlinear PDE systems and extends
Viscosity Solutions of Crandall-Ishii-Lions to the general vector case.
Key ingredient is the discovery of a notion of Extremum for maps which
extends min-max uniquely and allows for ``nonlinear passage of
derivatives" to test maps. The notions supports uniqueness, existence
and stability results, preserving most features of the scalar viscosity
counterpart. We will also discuss applications in vector-valued Calculus
of Variations in $L^\infty$ and Hamilton-Jacobi PDE with vector
solution.