With the arrival of the internet of things and the rise of wearable computing, electronics are playing an increasingly important role in our everyday lives. Until recently, however, the rigid angular nature of traditional electronics has prevented them from being integrated into many of the organic, curved shapes that interface with our bodies (such as ergonomic equipment or medical devices) or the natural world (such as aerodynamic or optical components). In the past few years, many groups working in advanced manufacturing and soft robotics have endeavored to develop strategies for fabricating electronics on these curved surfaces. This is their story. In this work, we describe the motivations, challenges, methodologies, and applications of curved electronics, and provide a outlook for this promising field.