  The fusion rings associated to affine Kac-Moody algebras appear in several
different contexts in math and mathematical physics. In this paper we find all
automorphisms of all affine fusion rings, or equivalently the symmetries of the
corresponding fusion coefficients. Most of these are directly related to
symmetries of the corresponding Coxeter-Dynkin diagram. We also find all pairs
of isomorphic affine fusion rings.
