What kinds of shapes can you make by folding a sheet of paper? How strong can you make them, or how flexible? Although we've been folding paper for centuries, we're still discovering fascinating new answers to these questions. Origami-inspired structures can improve the energy-efficiency of massive buildings, deliver drugs deep within the body, power spacecraft and even stop bullets. As we learn to manipulate sheets as thin as a single atom, humanity approaches the ultimate origami challenge--folding structures as rich and varied as those nature achieves through folding proteins. We will discuss how all of these structures are achieved by mastering the geometrical structure hidden within every sheet of paper.