In this paper we put forward a new time series model, which describes nonlinearity and seasonality simultaneously. We discuss its representation, estimation of the parameters and inference. This seasonal STAR (SEASTAR) model is examined for its practical usefulness by applying it to 18 quarterly industrial production series. The data are tested for smooth-transition nonlinearity and for time-varying seasonality. We find that the model fits the data well for 14 of the 18 series. We also consider out-of-sample forecasting where we compare forecasts from the SEASTAR models with forecasts from nested models. It turns out that the SEASTAR model sometimes outperforms the other models, particularly for large horizons. Finally, we compare the SEASTAR models with STAR models for the 14 corresponding seasonally adjusted series, and we find that the estimated business cycle chronologies can be markedly different.