Empirical research on returns distributions has been ongoing since the early 1960s: see, for example, the surveys in Kon (1984), Badrinath and Chatterjee (1988) and Mittnik and Rachev (1993a). These have almost universally found that such distributions are characterised by the ‘stylised facts’ of fat tails and high peakedness – excess kurtosis – and are often skewed. However, there have been several recent developments in statistics and econometrics that have led to considerable advances in the analysis of such distributions. To set the scene for subsequent analysis, section 1 presents initial descriptive analysis of the distributional properties of three typical return series, before section 2 reviews two of the most important theoretical models for examining return distributions, the stable process and, much more briefly since it was analysed in great detail in the previous chapter, the ARCH process. Section 3 generalises the discussion to consider tail shapes of distributions and methods of estimating indices of these shapes, while section 4 reviews existing empirical research and offers new evidence from our own returns series. Section 5 considers the implications of fat-tailed distributions for testing the conventional maintained assumption of time series models of returns, that of weak, or covariance, stationarity. Section 6 switches attention to modelling the central part of returns distributions and section 7 reviews data analytic methods of modelling skewness and kurtosis. The distributional properties of absolute returns are the focus of section 8, and a summary and some further extensions are provided in section 9.