This review considers the present state of understanding of the growth of carbon nanotubes based on TEM observations and numerical simulations. In the highlights of these experimental and theoretical approaches, various mechanisms, already proposed in the literature, for single and multi-shell nanotubes nucleation and growth are reviewed. A good understanding of nanotube growth at the atomic level is probably one of the main issues either to develop a nanotube mass production process or to control growth in order to obtain well-designed nanotube structures.