traditionally weak stationarity of a random field $\{x(t) : t\in \mathbf{t}\}$ over an index space $\mathbf{t}$ is defined with respect to a translation operation in $\mathbf{t}$. but this classical notion of stationarity does not extend to related random fields, as for example to the field of averages of $\{x(t): t\in \mathbf{t}\}$. in order to equip this latter field with a stationarity property one introduces a generalized translation in $\mathbf{t}$ which arises from a generalized convolution structure in the space $m^b(\mathbf{t})$ of bounded measures on $\mathbf{t}$. there are two fundamental constructions providing such (hypergroup) convolution structures on the index spaces $\mathbf{z}_+$ and $\mathbf{r}_+$, in terms of polynomial sequences and families of special functions, respectively.\\ in the present talk emphasis will be put on polynomially stationary random fields $\{x(n): n\in\mathbf{z}_+\}$ which were studied for the first time by r.~lasser and m.~leitner about 20 years ago. in the meantime the theory has developed interesting applications such as regularization, moving averages and prediction.\\ for square-integrable radial random fields over graphs, j.p.~arnaud has coined a notion of stationarity which yields spectral and karhunen type representations. these fields are related to polynomially stationary random fields over $\mathbf{z}_+$, where the underlying polynomial sequence generates the cartier-dunau convolution structure in $m^b(\mathbf{z}_+)$. an analogous approach related to special function stationarity of random fields over $\mathbf{r}_+$ seems promising, but requires further progress.