Single-wall carbon nanotubes (SWCNTs) represent an excellent example of materials by design with many of their outstanding properties predicted by theory prior to their synthesis. Both experimental and theoretical work on these novel nanowires continue to increase at a breathtaking pace. Herein we describe some of their fundamental properties on which much of this work is built. After discussing their structure and symmetries, we emphasize their exceptional electronic properties. The standard one-parameter graphene sheet model of SWCNTs, introduced in the earliest published paper on extended SWCNTs, is discussed in terms of both its successes and limitations. The strong interplay between theory and experiment that this area has enjoyed is also discussed. In addition, several opportunities for further study are touched upon.