In spite of its excellent electronic properties, the use of graphene in field-effect transistors is not practical at room temperature without modification of its intrinsically semimetallic nature to introduce a bandgap(1-4). Quantum confinement effects can create a bandgap in graphene nanoribbons, but existing nanoribbon fabrication methods are slow and often produce disordered edges that compromise electronic properties(2-4). Here, we demonstrate the self-organized growth of graphene nanoribbons on a templated silicon carbide substrate prepared using scalable photolithography and microelectronics processing. Direct nanoribbon growth avoids the need for damaging post-processing. Raman spectroscopy, high-resolution transmission electron microscopy and electrostatic force microscopy confirm that nanoribbons as narrow as 40 nm can be grown at specified positions on the substrate. Our prototype graphene devices exhibit quantum confinement at low temperatures (4 K), and an on-off ratio of 10 and carrier mobilities up to 2,700 cm(2) V(-1) s(-1) at room temperature. We demonstrate the scalability of this approach by fabricating 10,000 top-gated graphene transistors on a 0.24-cm(2) SiC chip, which is the largest density of graphene devices reported to date.