The problems posed by the synthesis and purification of single-walled carbon nanotubes (SWNTs) have inhibited progress in the field. In this article, we review the methods available for measuring the purity of SWNTs and the current status of processes designed to purify them.We emphasize the hierarchy of the purification steps that must be developed in order to obtain high-quality material suitable for the full range of advanced applications that are envisioned for the ultimate carbon nanofiber.We review two strategies for SWNT purification, the assessment of SWNT purity by use of near-IR spectroscopy and its application to the thermal oxidation of thin films of SWNTs, as well as recent advances in the separation of metallic and semiconducting SWNTs. While substantial progress has been made in the purification and separation of SWNTs, we note the need for quality control and quality assurance within the industry. Much work remains before pure SWNTs of specific lengths, diameters, and chirality can be made available for applications.