The organization of carbon nanotubes into well-defined straight or curved geometries and arrays on surfaces is a critical prerequisite for their integration into nanocircuits and a variety of functional nanosystems. We review the recent development of a new approach to carbon nanotube organization based on self-organized growth directed by well-defined crystal surfaces, or "nanotube epitaxy". We identify three different modes of surface-directed growth, namely by atomic rows, atomic steps, and nanofacets. Particular emphasis is given here to the combinations of such surface-directed growth with external forces—like those exerted by an electric field or gas flow—for the creation of well-defined complex geometries, including crossbar architectures, serpentines, and coils.