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We propose a novel framework for the statistical characterization of fading dispersive channels that do not satisfy the assumption of wide-sense stationary uncorrelated scattering (WSSUS). The local scattering function (LSF) and the channel correlation function (CCF) are introduced and shown to characterize, respectively, the mean power and the correlation of non-WSSUS scatterers. Furthermore, the practically important class of doubly underspread (DU) channels is introduced, and it is shown that for DU channels, the LSF has numerous useful properties. The practical relevance of our approach is illustrated via application examples and numerical experiments.