hypergroups are locally compact spaces for which the space of bounded measures can be made a banach algebra by introducing an axiomatically determined convolution. prominent constructions of such convolutions over $\mathbb {z}_+$ and $\mathbb {r}_+$ are performed via polynomials or special functions respectively. in order to establish canonical representations of convolution semigroups of probability measures on a commutative hypergroup $k$ (in the sense of schoenberg correspondence and levy-khintchine decomposition) one needs to study negative-definite functions on the dual $k$ $\hat{}$ of $k$ which in general is not a hypergroup. with appropriate definition of negative-definiteness on $k$ $\hat{}$ some harmonic analysis can be developed, and for large classes of (euclidean) hypergroups the structure of negative-definite functions and of the corresponding convolution semigroups can be illuminated.