Due to its higher degree of control and its scalability, catalytic chemical vapour deposition is now the prevailing synthesis method of carbon nanotubes. Catalytic chemical vapour deposition implies the catalytic conversion of a gaseous precursor into a solid material at the surface of reactive particles or of a continuous catalyst film acting as a template for the growing material. Significant progress has been made in the field of nanotube synthesis by this method although nanotube samples still generally suffer from a lack of structural control. This illustrates the fact that numerous aspects of the growth mechanism remain ill-understood. The first part of this review is dedicated to a summary of the general background useful for beginners in the field. This background relates to the carbon precursors, the catalyst nanoparticles, their interaction with carbonaceous compounds and their environment. The second part provides an updated review of the influence of the synthesis parameters on the features of nanotube samples: diameters, chirality, metal/semiconductor ratio, length, defect density and catalyst yield. The third part is devoted to important and still open questions, such as the mechanism of nanotube nucleation and the chiral selectivity, and to the hypotheses currently proposed to answer them.