This paper studies the behavior of cryptocurrencies’ financial time series, of which Bitcoin is the most prominent example. The dynamics of these series are quite complex, displaying extreme observations, asymmetries, and several nonlinear characteristics that are difficult to model and forecast. We develop a new dynamic model that is able to account for long memory and asymmetries in the volatility process, as well as for the presence of time-varying skewness and kurtosis. The empirical application, carried out on 606 cryptocurrencies, indicates that a robust filter for the volatility of cryptocurrencies is strongly required. Forecasting results show that the inclusion of time-varying skewness systematically improves volatility, density, and quantile predictions at different horizons.