In most two-dimensional nonlinear shell theories, such as that proposed by W.T Koiter, the stored energy function is a simple expression of the first and second fundamental forms of the unknown deformed middle surface of the shell.This observation suggests an alternate approach to nonlinear shell theory, where the fundamental forms would be regarded as the primary unknowns, instead of the customary displacement field of the middle surface.Such an approach yields a constrained minimization problem, where the unknowns must naturally satisfy, possibly in a weak sense, the classical Gauss and Codazzi-Mainardi equations.This approach, which often bears the name of "intrinsic theories for shells" in the Engineering and Computational Mechanics circles, presents the advantage of directly providing the stresses inside a shell.The aim of this talk is to review some recent significant steps toward a mathematical justifications of this approach, such as: -a thorough study of the continuity of a surface as a function of its fundamental forms for various natural topologies;-a proof of a nonlinear Korn inequality "on a surface";-a rigorous mathematical analysis of intrinsic theories for linearly elastic shells, in which case the new unknowns are the components of the linearized change of metric and linearized change of curvature tensors.