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Fullerenes are the most exciting new set of chemical structures discovered in the last part of the 20th century. The advent of these novel structures has led to fascinating new scientific questions and to intriguing potential applications. This article provides an overview of the structures of the common members of the fullerene class of carbon allotropes and introduces the essentials of fullerene reactivity. Specific types of reactivity discussed herein include reactions of fullerenes with nucleophiles and electrophiles, oxidations and reductions, as well as reactions with metals and cycloaddition reactions. A glimpse of the potential applications of these fascinating molecules is also provided. The focus will be on the chemistry of C60 as it is the most available and the best-understood member of an expanding set of chemical structures. The chemistry of C70 is discussed briefly as well, demonstrating how strain directs the reactivity of this less-symmetrical structure.