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Many-electron effects often dramatically modify the properties of reduced dimensional systems. We report calculations, based on an ab initio many-electron Green’s function approach, of electron-hole interaction effects on the optical spectra of small-diameter single-walled carbon nanotubes. Excitonic effects qualitatively alter the optical spectra of both semiconducting and metallic tubes. Excitons are bound by in the semiconducting (8,0) tube and by in the metallic (3,3) tube. These large many-electron effects explain the discrepancies between previous theories and experiments.