Electronic devices have advanced from their heavy, bulky origins to become smart, mobile appliances. Nevertheless, they remain rigid, which precludes their intimate integration into everyday life. Flexible, textile and stretchable electronics are emerging research areas and may yield mainstream technologies1,2,3. Rollable and unbreakable backplanes with amorphous silicon field-effect transistors on steel substrates only 3 μm thick have been demonstrated4. On polymer substrates, bending radii of 0.1 mm have been achieved in flexible electronic devices5,6,7. Concurrently, the need for compliant electronics that can not only be flexed but also conform to three-dimensional shapes has emerged3. Approaches include the transfer of ultrathin polyimide layers encapsulating silicon CMOS circuits onto pre-stretched elastomers8, the use of conductive elastomers integrated with organic field-effect transistors (OFETs) on polyimide islands9, and fabrication of OFETs and gold interconnects on elastic substrates10 to realize pressure, temperature and optical sensors11,12,13,14. Here we present a platform that makes electronics both virtually unbreakable4 and imperceptible. Fabricated directly on ultrathin (1 μm) polymer foils, our electronic circuits are light (3 g m−2) and ultraflexible and conform to their ambient, dynamic environment. Organic transistors with an ultra-dense oxide gate dielectric a few nanometres thick formed at room temperature enable sophisticated large-area electronic foils with unprecedented mechanical and environmental stability: they withstand repeated bending to radii of 5 μm and less, can be crumpled like paper, accommodate stretching up to 230% on prestrained elastomers, and can be operated at high temperatures and in aqueous environments. Because manufacturing costs of organic electronics are potentially low, imperceptible electronic foils may be as common in the future as plastic wrap is today. Applications include matrix-addressed tactile sensor foils for health care and monitoring, thin-film heaters, temperature and infrared sensors, displays15, and organic solar cells16.