A time series model in which the signal is buried in noise that is non-Gaussian may throw up observations that, when judged by the Gaussian yardstick, are outliers. We describe an observation driven model, based on an exponential generalized beta distribution of the second kind (EGB2), in which the signal is a linear function of past values of the score of the conditional distribution. This specification produces a model that is not only easy to implement, but which also facilitates the development of a comprehensive and relatively straight forward theory for the asymptotic distribution of the maximum likelihood estimator. Score driven models of this kind can also be based on conditional t-distributions, but whereas these models carry out what, in the robustness literature, is called a soft form of trimming, the EGB2 distribution leads to a soft form of Winsorizing. An EGARCH model based on the EGB2 distribution is also developed. This model complements the score driven EGARCH model with a conditional t-distribution. Finally dynamic location and scale models are combined and applied to data on the UK rate of inflation.