Although many of the models commonly used in empirical finance are linear, the nature of financial data suggests that non-linear models are more appropriate for forecasting and accurately describing returns and volatility. The enormous number of non-linear time series models appropriate for modeling and forecasting economic time series models makes choosing the best model for a particular application daunting. This classroom-tested advanced undergraduate and graduate textbook, first published in 2000, provides a rigorous treatment of recently developed non-linear models, including regime-switching and artificial neural networks. The focus is on the potential applicability for describing and forecasting financial asset returns and their associated volatility. The models are analysed in detail and are not treated as 'black boxes'. Illustrated using a wide range of financial data, drawn from sources including the financial markets of Tokyo, London and Frankfurt.