One of the most prominent aspects of the novel form of carbon molecules, the ''fullerenes'', is the characteristic features of the C-C bond networks. The fullerene hollow closed cage consists of five- and six-membered rings, providing a spherical surface with no adjacent pentagons in the network (isolated pentagon rule, IPR). Here we describe a novel picture of growth of fullerenes, paying special attention to the initial stage of growth of five- and six-membered ring networks. Some important precursors associated with specific fullerenes are suggested in connection with the structure and stability of small carbon clusters.