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.