The theory of thin elastic surfaces is a source of many fascinating problems in the calculus of variations. When an elastic surface is deformed, its energy can be roughly decomposed into two parts – a "stretching/membrane energy" that is non-convex in the derivative of the deformation map and a "bending energy" that depends on higher order derivatives of the deformation map. The interplay between these two terms gives rise to a variety of interesting phenomena unique to thin elastic objects. We are in part motivated by wrinkling observed in highly stretched polymer sheets. This thesis has three parts: In Chapter 3, we consider a wide class of models known as "Cosserat shells" and identify a new, physically meaningful convexity condition that leads to the existence of energy minimizers for these models. We argue that this convexity condition is suitable for predicting wrinkling phenomena. In Chapter 4, we focus on a pure membrane version of the model in Chapter 3, which in general is not even rank-one convex. Nevertheless, we prove that it admits energy minimizers when the image surface is constrained to lie on some prescribed embedded oriented surface in $\mathbb{R}^3$. Under additional assumptions, we show that the minimizers are homeomorphisms onto their image and are weak solutions to the spatial equilibrium equations. In both Chapter 3 and Chapter 4, we ensure that the minimizers are locally injective/orientation preserving by requiring the energy density function to grow unboundedly as an appropriate notion of the local (signed) area/volume measure approaches zero. In Chapter 5, we study the membrane energy from the viewpoint of 3D to 2D dimension reduction. We embed our 3D variational problems into an appropriate class of parametrized measures and obtain a compactness result as the thickness of the body goes to zero. The putative membrane limit is defined on a class of