It is argued that the growth mechanism of single-wall carbon nanotubes (SWNTs) is a kind of solid-liquid-solid (SLS) catalytic graphitization of non-graphitic forms of carbon (predominantly amorphous carbon, condensed at early stages of the carbon vapor relaxation) catalyzed by molten supersaturated carbon-metal nanoparticles. The liquefaction of nanoparticles of the catalytic metals in contact with the amorphous carbon below the equilibrium eutectic temperature of the corresponding metal-carbon alloys is considered as a decisive condition for the SWNT formation both at the nucleation and at the growth stages. It leads to a pronounced increment of carbon solubility (up to 50 at.% C), which alters the character of the interaction of the precipitated graphitic form of carbon with the catalyst surface and presumably can result in the SWNT nucleation. Furthermore, it lowers the energy barrier between the initial amorphous and final graphitic carbon phases. The experimental dependencies of the SWNT abundance in the soot on the synthesis temperature, pressure, carrier gas flow velocity, and the initial catalyst content in the target support the idea of the a "catalyst particle size window" in which the supersaturated metal-carbon alloy may remain liquid below the equilibrium eutectic temperature. (C) 2002 Elsevier Science B.V. All rights reserved.