The speaker is dealing with commutative hypergroups whose base space is a compact subset $K$ of the $k$-dimensional Euclidean space and whose convolution in the set $M(K)$ of bounded measures on $K$ is defined via sequences of $k$-variable polynomials on $K$. Examples of such hypergroups are the unit interval, the closed unit square, the disk, the parabolic biangle, and the simplex. \vskip .1in \noindent There is an elaborate harmonic analysis available for compact commutative hypergroups based on the generalized translation operation in $M(K)$. Significant results have been established in analogy to but remarkably distinct from the classical framework of a compact Abelian group. We just note that in general there is no dual hypergroup attached to the given hypergroup $K$, that the Plancherel measure is rarely full, and that positive definite functions can be unbounded. Nevertheless, one has a Haar measure for $K$ and an extended Fourier-Stieltjes theory. \vskip .1in \noindent Applications of commutative hypergroups point in various directions: Hecke algebras, second order differential operators, and limit theorems of probability theory are just three of them. In the talk to be given the speaker restricts himself to cone-embedded polynomial hypergroups $K$ and describes canonical representations of processes with independent increments in $K$.