Tremendous progress has been achieved during the past 20 years on not only improving the yields of carbon nanotubes and move progressively towards their mass production, but also on gaining a profound fundamental understanding of the nucleation and the growth processes. Parameters that influence the yield but also the quality (e.g., microstructure, homogeneity within a batch) are better understood. The influence of the carbon precursor, the reaction conditions, the presence of a catalyst, the chemical and physical status of the latter, and other factors have been extensively studied. The purpose of the present Review is not to list all the experiments reported in the literature, but rather to identify trends and provide a comprehensive summary on the role of selected parameters. The role of the catalyst occupies a central place in this Review as a careful control of the metal particle size, particle dispersion on the support, the metastable phase formed under reaction conditions, its possible reconstruction, and faceting strongly influence the diameter of the carbon nanotubes, their structure (number of walls, graphene sheet orientation, chirality), their alignment, and the yield. The identified trends will be compared with recent observations on the growth of graphene. Recent results on metal-free catalysts will be analyzed from a different perspective.