The remarkable structure and properties of carbon nanotubes will be reviewed briefly. It will then be shown how the vibrational spectra of one tiny tube, only about 1nm (10Å) in diameter, can be observed experimentally. Whereas Raman spectroscopy normally measures vibrational frequencies, under resonant conditions the spectra not only provide important information about the electronic structure through the strong coupling between electrons and lattice vibrations, but for this one-dimensional system, resonance Raman spectroscopy at the single nanotube level also determines the geometrical structure of the resonant nanotube. The implications of these unusual properties are discussed. The use of Raman spectroscopy to characterize carbon nanotubes for various applications will be discussed, with particular reference given to the procedure developed at the University of Connecticut for separating semiconducting and metallic nanotubes.