  In the present paper, we propose a new discrete surface theory on 3-valent
embedded graphs in the 3-dimensional Euclidean space which are not necessarily
discretization or approximation of smooth surfaces. The Gauss curvature and the
mean curvature of discrete surfaces are defined which satisfy properties
corresponding to the classical surface theory. We also discuss the convergence
of a family of subdivided discrete surfaces of a given 3-valent discrete
surface by using the Goldberg-Coxeter construction. Although discrete surfaces
in general have no corresponding smooth surfaces, we may find one as the limit.
