  We consider a general theory of curvatures of discrete surfaces equipped with
edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces
are derived from the faces' areas and mixed areas. Remarkably these notions are
capable of unifying notable previously defined classes of surfaces, such as
discrete isothermic minimal surfaces and surfaces of constant mean curvature.
We discuss various types of natural Gauss images, the existence of principal
curvatures, constant curvature surfaces, Christoffel duality, Koenigs nets,
contact element nets, s-isothermic nets, and interesting special cases such as
discrete Delaunay surfaces derived from elliptic billiards.
