The discovery of C₆₀ and C₇₀, icosahedral spherical and ellipsoidal carbon species in September 1985, followed by their successful synthesis in 1990, excited the imagination of many scientists and demanded a radical revision of old, seemingly well-founded preconceptions in carbon science, leading to the institution of a new, multidisciplinary science of chemistry, physics and materials science `in the round'. New, unique carbon materials have been discovered, like the nanotubes, buckyonions and endohedral metallofullerenes; and the fullerenes themselves, and their derivatives, have been found to possess a plethora of interesting properties, ranging from the inhibition of HIV-I protease to superconductivity and ferromagnetism. New and exciting material aplenty, all of which is here expertly reviewed by the founders of the science, as well as its most expert practitioners.