A scanning tunneling microscope (STM) was used to explore the local electrical characteristics of single-wall carbon nanotubes. As the STM tip was moved along the length oi the nanotubes, well-defined positions were found where the transport current changes abruptly from a graphitic-like response to one that is highly nonlinear and asymmetrical, including near-perfect rectification. The observations are consistent with the existence of localized, on-tube nanodevices of a type proposed theoretically.