Non-Gaussian translation processes is a method used primarily in engineering to model non-Gaussian stochastic processes. With a strong connection to copulas, this methodology separates the correlation structure of a stochastic process from the marginal distributions of the specific data points, allowing for flexibility in user selection of distributional characteristics. When applied in a multivariate setting to space-time models, the result is a non-linear spatio-temporal dynamic model, that can be fit using an extended Kalman filter. It can be combined with a number of existing parameterizations of linear dynamic model structures allowing these existing methods to be easily extended to include non-Gaussian marginals. The methodology is described along with theoretical properties of the resulting process. It is then applied in an extreme value setting using stable laws as marginal distributions.