A theoretical description of electron spin resonance (ESR) in 1D interacting metals is given, with primary emphasis on carbon nanotubes. The spin-orbit coupling is derived, and the resulting ESR spectrum is analyzed using a low-energy field theory. Drastic differences in the ESR spectra of single-wall and multiwall nanotubes are found. For single-wall tubes, the predicted double peak spectrum is linked to spin-charge separation. For multiwall tubes, a single narrow asymmetric peak is expected.