Diffusion of carbon is the crucial step in chemical vapor deposition (CVD) growth of carbon nanotubes (CNTs) On metal nanoparticles. Using first principles techniques we model the binding and diffusion of carbon atoms and dimers on late transition (Ni, Pd, Pt) and coinage (Cu, Ag, Au) metal catalysts. We consider various diffusion mechanisms including both surface and subsurface channels, finding the lowest activation barriers for carbon adatoms on nanoparticles of coinage metals. For these metals, our calculations further show that the diffusion is restricted to the nanoparticle surface when diatomic carbon is initially obtained from the decomposition of the precursor gas. In combination with the results for the binding of CNT fragments to metal nanoparticles, we conclude that coinage metal. catalysts, in particular Cu, favor CVD growth of CNTs at low temperatures and with narrow chirality distributions.