A review is presented of the resonance Raman spectra from individual isolated single-wall carbon nanotubes (SWNTs). A brief summary is given of how the measurements are made. Why the resonance Raman effect allows single-carbon nanotube spectra to be observed easily and under normal operating conditions is summarized. The important structural information that is provided by single-nanotube spectroscopy using one laser line is discussed, and what else can be learned from tunable laser experiments is reviewed. Particular attention is given to the determination of the nanotube diameter and of the energy of its van Hove singularities Eii. Applications of single-nanotube spectroscopy are emphasized, such as measurements of isolated SWNTs connected with circuit-based samples and of isolated SWNTs mounted on an atomic force microscope tip. A critical assessment of the opportunities and limitations of the resonance Raman method for structural (n, m) identification is presented. The trigonal warping effect, which is central to the (n, m) identification in resonance Raman spectroscopy, is discussed in simple terms, and the importance of this effect in nanotube science and applications is reviewed.