In this talk, we will consider a class of models for highly deformable elastic surfaces that account for both curvature and nonlinear membranal strains. We assume a ``polyconvexity'' condition for the energy density that reduces to a physically correct membrane model in the absence of bending. With appropriate growth conditions, we establish the existence of energy minimizers. The orientation of a minimizing configuration is preserved via the blowup of the energy density as a version of the local volume ratio approaches zero. We then specialize our results to three constrained versions of the theory commonly employed in the subject. This is joint work with Timothy Healey.