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Origami, the art of folding paper into intricate shapes, has growing practical applications across fields such as architectural design, therapeutics, deployable space structures, antenna design, and soft robotics. One promising yet largely unexplored area is curved tile origami, which can store elastic energy, offering opportunities to develop next-generation functional materials, structures, and actuators. In this talk I will present a general theory of curved origami and systematic design methods for constructing large-scale, complex curved origami structures. Additionally, I will present methods to accurately calculate the stored elastic energy and the folding motions of curved origami, and I will illustrate my theoretical results by presenting some complex structures I have folded. This theory has inspired applications of curved origami in fields ranging from medical devices to a vertical-axis wind turbine, to architected materials.