The growth and growth rates of single-crystal nanowires have been studied. Extensive theoretical calculations have been performed. The growths by the vapor-phase mechanisms have been considered. These mechanisms include the vapor-liquid-solid (VLS), vapor-solid-solid, oxide-assisted growth, and the self-catalytic growth mechanisms. The modeling for nanowire growth and growth rate takes adsorption, desorption, surface scattering, and diffusion into account. The fundamentals underlying the growth rates and the parameters dictating them have been elucidated. The role of foreign element catalytic agents in the VLS growth has been examined. Experimental evidences have been advanced to quantify the influence of these parameters. Dependence of nanowire growth rates on temperature, nanowire radius, and chamber pressure has been studied. These growth rates obtained by theoretical, empirical, and experimental techniques compare well. The study solves important scientific problems, conflicts, controversies, and anomalies pertaining to nanowire growth. It uncovers basic processes underlying the controversies. It explains even the intricate details of the fundamentals governing the nanowire growths and growth rates. It elucidates why the nanowire growth rate by the molecular beam epitaxy is very low. Remarkably, it manifests incredibly tiny peaks in very thin nanowires observed experimentally several decades ago and explains the origin of these tiny peaks.