\indent The present lecture is devoted to recent developments on random walks with spherical symmetry, a topic which was opened to research by J.F.C. Kingman in 1963, and which has developed wide-ranging applications through the work of W. Hazod, M. Rösler, and M. Voit. The analytic method to be described in the talk concerns generalized convolutions of measures on hypergroups, in particular on the self-dual commutative hypergroup of positive semidefinite (Hermitian) matrices. These hypergroups are defined via Bessel functions of higher rank. A typical application of the hypergroup setting is the study of Bessel random walks on matrix cones and their convergence to Wishart distributions.