Multiple-scattering theory, as it is applicable to metallic systems, is generalized systematically compared with the usual formulation. The generalization refers to the reference system, which conventionally is free space, but which can be chosen to be much more general. For example, in describing a binary alloy, a defect in a metal, or a finite impurity-cluster model for the latter system, the reference system may be an average medium, the unperturbed metallic host system, or the finite host cluster, respectively. Applications are indicated for the generalized Friedel sum rule, the local density of states, and the transition probability of a Bloch electron. A state in a system is shown to be built up of properly generalized incoming and outgoing waves.