The chiral structure of single-walled carbon nanotubes (SWNTs) and the edge structure of graphene nanoribbons (GNRs) play an important role in determining their electronic and phonon structures. Spectroscopic methods, which require simple sample preparation and cause minimal sample damage, are the most commonly utilized techniques for determining the structures of SWNTs and graphene. In this review the current status of various spectroscopic methods are presented in detail, including resonance Raman, photoluminescence (PL), and Rayleigh scattering spectroscopies, for determination of the chiral structure of individual SWNTs and the edge structure of isolated graphene, especially of graphene nanoribbons. The different photophysical processes involved in each spectroscopic method are reviewed to achieve a comprehensive understanding of the electronic and phonon properties of SWNTs and graphene. The advantages and limitations of each spectroscopic method as well as the challenges in this area are discussed.