This book deals with the empirical analysis of financial time series with an explicit focus on, first, describing the data in order to obtain insights into their dynamic patterns and, second, out-of-sample forecasting. We restrict attention to modelling and forecasting the conditional mean and the conditional variance of such series – or, in other words, the return and risk of financial assets. As documented in detail below, financial time series display typical nonlinear characteristics. Important examples of those features are the occasional presence of (sequences of) aberrant observations and the plausible existence of regimes within which returns and volatility display different dynamic behaviour. We therefore choose to consider only nonlinear models in substantial detail, in contrast to Mills (1999), where linear models are also considered. Financial theory does not provide many motivations for nonlinear models, but we believe that the data themselves are quite informative. Through an extensive forecasting experiment (for a range of daily and weekly data on stock markets and exchange rates) in chapter 2, we also demonstrate that linear time series models simply do not yield reliable forecasts. Of course, this does not automatically imply that nonlinear time series models might, but it is worth a try. As there is a host of possible nonlinear time series models, we review only what we believe are currently the most relevant ones and the ones we think are most likely to persist as practical descriptive and forecasting devices.