This talk will report on studies of the properties of carbon nanotubes of known chiral index, as determined by Rayleigh scattering spectroscopy. These properties include the mechanical stiffness, the electromechanical response, and basic electrical transport properties. The behavior of heterojunctions between nanotubes of different chiral indices will also be described. We have measured the mechanical properties of suspended graphene sheets, which show ultrahigh stiffness and the highest tensile strength ever measured. Finally, I will describe recent work on the use of graphene for high-frequency resonators.