Despite recent progress in producing transparent and bendable thin-film transistors using graphene and carbon nanotubes(1,2), the development of stretchable devices remains limited either by fragile inorganic oxides or polymer dielectrics with high leakage current(3,4). Here we report the fabrication of highly stretchable and transparent field-effect transistors combining graphene/single-walled carbon nanotube (SWCNT) electrodes and a SWCNT-network channel with a geometrically wrinkled inorganic dielectric layer. The wrinkled Al2O3 layer contained effective built-in air gaps with a small gate leakage current of 10(-13) A. The resulting devices exhibited an excellent on/off ratio of similar to 10(5), a high mobility of similar to 40 cm(2) V-1 s(-1) and a low operating voltage of less than 1V. Importantly, because of the wrinkled dielectric layer, the transistors retained performance under strains as high as 20% without appreciable leakage current increases or physical degradation. No significant performance loss was observed after stretching and releasing the devices for over 1,000 times. The sustainability and performance advances demonstrated here are promising for the adoption of stretchable electronics in a wide variety of future applications.