The use of Raman spectroscopy to reveal the remarkable structure and the unusual electronic and phonon properties of single wall carbon nanotubes (SWNTs) is reviewed comprehensively. The various types of Raman scattering processes relevant to carbon nanotubes are reviewed, and the theoretical foundations for these topics are presented. The most common experimental techniques used to probe carbon nanotubes are summarized, followed by a review of the novel experimental findings for each of the features in the first order and second order Raman spectra for single wall carbon nanotubes. These results are presented and discussed in connection with theoretical considerations. Raman spectra for bundles of SWNTs, for SWNTs surrounded by various common wrapping agents, and for isolated SWNTs at the single nanotube level are reviewed. Some of the current research challenges facing the field are briefly summarized.