Driven by the potential application to wireless communications, intensive research efforts have been made on the study of various selective combining (SC) schemes in the past decade. Nevertheless, regardless of its practical importance, performance analysis of multi-branch SC over spatially correlated fading channels is not available in literature except for the simplest case of dual diversity. The major difficulty lies in the fact that SC has its root in the theory of order statistics, and yet systematic methodology has been developed mainly for order statistics obtained from an independent population. In this paper, we formulate the problem in a very general framework, whereby a general solution is derived. The application of the new solution to SC with different modulation schemes and operational environments is elaborated. The fading environments to be addressed include correlated Nakagami, Rayleigh, and Rician channels. Numerical examples are also presented to illustrate the use of the theory.