Biology is curved, soft and elastic; silicon wafers are not. Semiconductor technologies that can bridge this gap in form and mechanics will create new opportunities in devices that require intimate integration with the human body. This paper provides an overview of ideas for electronics, sensors and actuators that offer the performance of state-of-the-art, wafer-based systems but with the mechanical properties of a rubber band. We explain the underlying materials science and mechanics of these approaches, and illustrate their use in bio-integrated, `tissue-like' devices with unique diagnostic and therapeutic capabilities, when conformally laminated onto the heart, brain or skin. Demonstrations in animal models and in humans illustrate the diverse functionality offered by these technologies, and suggest several clinically relevant applications.