Fullerenes are all-carbon polyhedral cage molecules in which each atom is directly bonded to three neighbours and the n atoms and 3n/2 edges are arranged in 12 pentagonal and n/2–10 hexagonal rings. Simple mathematical and physical models can be used to count and construct isomers, to assign symmetries, to estimate geometries, to predict the (rare) appearances of property closed πelectron configurations and to rationalize trends in overall stability in this new and growing family of molecules.