Using first-principles calculations, we model the chemical vapor deposition (CVD) growth of carbon nanotubes (CNT) on nanoparticles of late-transition (Ni, Pd, Pt) and coinage (Cu, Ag, Au) metals. The process is analyzed in terms of the binding of mono- and diatomic carbon species, their diffusion pathways, and the stability of the growing CNT. We find that the diffusion pathways can be controlled by the choice of the catalyst and the carbon precursor. Binding of the CNT through armchair edges is more favorable than through zigzag ones, but the relative stability varies significantly among the metals. Coinage metals, in particular Cu, are found to favor CVD growth of CNTs at low temperatures and with narrow chirality distributions.