The diverse applications of terahertz (THz) radiation and its importance to fundamental science makes finding ways to generate, manipulate and detect THz radiation one of the key areas of modern applied physics. One approach is to utilize carbon nanomaterials, in particular, single-wall carbon nanotubes and graphene. Their novel optical and electronic properties offer much promise to the field of THz science and technology. This article describes the past, current, and future of THz science and technology of carbon nanotubes and graphene. We will review fundamental studies such as THz dynamic conductivity, THz nonlinearities and ultrafast carrier dynamics as well as THz applications such as THz sources, detectors, modulators, antennas and polarizers.