Macroscopic fibers containing only Carbon NanoTubes (CNTs) will yield great advances in high-tech applications if they can attain a significant portion of the extraordinary mechanical and electrical properties of individual CNTs. Doing so will require that the CNTs in the fiber are sufficiently long, highly aligned and packed in an arrangement that is nearly free of defects. Here we review and compare the various methods for processing CNTs into neat fibers. These techniques may be divided into 'liquid' methods, where CNTs are dispersed into a liquid and solution-spun into fibers, and 'solid' methods, where CNTs are directly spun into ropes or yarns. Currently, these processes yield fibers whose properties are not sufficiently close to optimal; however, the last five years have seen rapid progress, and the production of commercially useful CNT fibers may be achieved in the next few years.