For a Gelfand pair $(G,H)$ consisting of a locally compact group $G$ and a compact subgroup $H$ of $G$, the $H$-bi-invariant measures on $G$ can be identified with measures on the double coset space $G/\!/H$. The canonical projection from $G$ onto $G/\!/H$ induces a convolution of measures on $G/\!/H$ such that $G/\!/H$ becomes a commutative hypergroup. Central to the present talk will be the discussion of L\'evy processes with values in the hypergroup $G/\!/H$ and their characterization in terms of complex-valued martingales. In order to achieve this goal some harmonic analysis on hypergroups has to be developed: the convolution hemigroups associated with L\'evy processes will be studied with the help of a Fourier transform. On Sturm-Liouville hypergroups the characterization of Gaussian hemigroups is obtained via martingales involving moments. For convolution semigroups, i.e.\ stationary L\'evy processess, the results yield characterizations of radial Brownian motions on Euclidean and hyperbolic spaces.