We describe a generalization of the Lefschetz fixed-point formula. The formula equates two invariants of a smooth, G-equivariant self-correspondence of a smooth compact manifold, where G is a compact group. As in the classical formula, one of our invariants is local and geometric and is based on a self-intersection construction, and the other is global and homological, and depends, roughly speaking, only on the R(G)-module trace of the R(G)-module map on equivariant K-theory induced by the correspondence.