I will talk about two problems from the physics of thin, elastic sheets which can be understood from a purely geometrical perspective. Some plants develop an ability to perform rapid motions (Venus flytraps, exploding Jewelweed seeds, etc.) even without muscles. They achieve this by manipulating the curvature of tissues, taking advantage of the relationship between strains and geometry to trigger rapid conformational changes from one state to another. I will discuss a new theoretical framework and related experiments for designing snapping structures by taking advantage of ideas from origami. In the second problem, I will develop a framework to understand how flat sheets can conform themselves to curved surfaces even without appreciable strains, in contradiction naive intuition that a flat sheet cannot wrap a spherical droplet without stretching. I will describe a geometric framework to understand the shapes of sheet-wrapped droplets which agrees well with experiments.