Smoothing splines have become an established method for fitting curves with a flexible but smooth form that is determined by the data. This is particularly useful where there is no a priori form for response to an underlying covariate, or where examination of the data indicates that standard linear or non-linear models are unlikely to give a good fit. In the context of longitudinal data, smoothing splines have proved useful in providing a quantitative description of treatment profiles across time. Comparison of treatment profiles then requires a good model for covariation between subjects and across measurements within

subjects. The embedding of smoothing splines within the linear mixed-model framework allows an easy route to the joint modeling of mean and variance via residual or restricted maximum likelihood (REML) estimation, and this chapter discusses these methods of modeling longitudinal data. For simplicity, we assume initially that data can be considered to be normally distributed (possibly after transformation), and later discuss extensions to non-normal data.