  In this paper we are interested in defining affine structures on discrete
quadrangular surfaces of the affine three-space. We introduce, in a
constructive way, two classes of such surfaces, called respectively indefinite
and definite surfaces. The underlying meshes for indefinite surfaces are
asymptotic nets satisfying a non-degeneracy condition, while the underlying
meshes for definite surfaces are non-degenerate conjugate nets satisfying a
certain natural condition. In both cases we associate to any of these nets
several discrete affine invariant quantities: a metric, a normal and a
co-normal vector fields, and a mean curvature. Moreover, we derive structural
and compatibility equations which are shown to be necessary and sufficient
conditions for the existence of a discrete quadrangular surface with a given
affine structure.
