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.