Single wall carbon nanotubes (SWCNT) have become a strategic material in the area of nanotechnologies nowadays, and catalytic chemical vapor deposition seems to be the most promising technique in view of an industrial-scale production. However, the selective catalytic production of single wall carbon nanotubes is still a challenge, since catalytic systems performances both in terms of selectivity and activity are still relatively low. One of the main challenges for the catalytic growth of SWCNT is the control of the catalyst nanoparticles size distribution along the high temperatures required by the process. This article provides a comprehensive overview of the state of the art of the strategies that have been followed to selectively grow single wall carbon nanotubes. It focuses on catalysts preparation and activity/selectivity and on the growth mechanism of these nanostructures. Particular attention is given to the identification of the parameters that control the selectivity of the reaction, such as the choice of the metal/support couple, the particle's sizes, and the chemical vapor deposition conditions.