Production methods for carbon nanotubes are now well established and allow their synthesis on a scale of grams per day. For many potential applications of this unique material, its purification still remains a crucial problem. In this article various purification methods for single- and multi-wall carbon nanotubes are reviewed. These methods are compared in terms of their capacity, efficiency, and effects on the tubes. In addition, the use of Raman spectroscopy for monitoring the chromatographic purification of single-wall nanotubes is described.