We demonstrate high electron mobility single-crystal InAs mesas monolithically integrated on amorphous dielectric substrates at a growth temperature of 300 °C. Critically, a room temperature mobility of ∼5800 cm 2 /(V s) was measured, the highest mobility reported for any thin-film semiconductor material system directly grown on a nonepitaxial substrate. Detailed modeling of the scattering mechanisms in the grown material indicates that the mobility is limited by surface roughness scattering, not the intrinsic material quality. We project that reducing the RMS surface roughness of the InAs from 1.8 to 1 nm would produce materials with room temperature mobilities of >10000 cm 2 /(V s), and RMS roughness of 0.5 nm would result in mobility of ∼20000 cm 2 /(V s), essentially identical with epitaxially grown materials. These results pave the way for growth of high-mobility materials directly onto the back end of silicon CMOS wafers and other nonepitaxial substrates such as glass, as well as polymers for flexible electronics.