In this paper we study new nonlinear GARCH models mainly designed for time series with highly persistent volatility. For such series, conventional GARCH models have often proved unsatisfactory because they tend to exaggerate volatility persistence and exhibit poor forecasting ability. Our main emphasis is on models that are similar to previously introduced smooth transition GARCH models except for the novel feature that a lagged value of conditional variance is used as the transition variable. This choice of the transition variable corresponds to the idea that high persistence in conditional variance is related to relatively infrequent changes in regime. U sing the theory of Markov chains we provide sufficient conditions for the stationarity and existence of moments of the considered smooth transition GARCH models and even some more general nonlinear GARCH models. Empirical applications to two exchange rate return series show that the new models can be superior to conventional GARCH models especially in longer term forecasting.