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Recent advances in mechanics and materials provide routes to integrated circuits that can offer the electrical properties of conventional, rigid wafer-based technologies but with the ability to be stretched, compressed, twisted, bent, and deformed into arbitrary shapes. Inorganic and organic electronic materials in microstructured and nanostructured forms, intimately integrated with elastomeric substrates, offer particularly attractive characteristics, with realistic pathways to sophisticated embodiments. Here, we review these strategies and describe applications of them in systems ranging from electronic eyeball cameras to deformable light-emitting displays. We conclude with some perspectives on routes to commercialization, new device opportunities, and remaining challenges for research.