We have developed a simple approach to high-performance, stretchable, and foldable integrated circuits. The systems integrate inorganic electronic materials, including aligned arrays of nanoribbons of single crystalline silicon, with ultrathin plastic and elastomeric substrates. The designs combine multilayer neutral mechanical plane layouts and "wavy" structural configurations in silicon complementary logic gates, ring oscillators, and differential amplifiers. We performed three-dimensional analytical and computational modeling of the mechanics and the electronic behaviors of these integrated circuits. Collectively, the results represent routes to devices, such as personal health monitors and other biomedical devices, that require extreme mechanical deformations during installation/use and electronic properties approaching those of conventional systems built on brittle semiconductor wafers.