AbstractThe main theorem in this paper is a far‐reaching generalization of Gleason's theorem on the weight enumerators of codes which applies to arbitrary‐genus weight enumerators of self‐dual codes defined over a large class of finite rings and modules. The proof of the theorem uses a categorical approach, and will be the subject of a forthcoming book. However, the theorem can be stated and applied without using category theory, and we illustrate it here by applying it to generalized doubly‐even codes over fields of characteristic 2, doubly‐even codes over ℤ/2fℤ, and self‐dual codes over the noncommutative ring 𝔽q + 𝔽q u, where u2 = 0. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)