Disclosed are methods of purifying mixtures comprising nanofibers and/or nanotubes and residual catalyst particles that are covered by outer layers of the nanotube or nanofiber material. The mixtures are exposed to electromagnetic radiation, which induces localized heating in the residual catalyst particles. The localized heating creates breaches in the outer layers. Thereafter, the residual catalyst particles may be removed under relatively mild conditions that do not significantly affect the structural integrity of the nanotubes or nanofibers. The methods of the invention have been used to particular advantage in the purification of single wall carbon nanotubes (SWNTs) synthesized using metal catalysts. For these SWNTs, microwave radiation is preferably used to induce the localized heating, the outer layers are preferably removed at least in part by carrying out the localized heating under air, and the residual catalyst may be removed by exposure to relatively dilute aqueous acid.