I will discuss models for longitudinal data, where the data consists of noisy measurements taken at several different time points for each individual, and the aim is to model how each individual's underlying response varies over time. If we assume linear variation of the responses over time, we could use a linear mixed model for this task. In this talk, I will discuss more flexible modelling approaches, which allow the variation of the response over time to be any smooth curve. There is a strong link with models for functional data, and I will describe previous work on adapting methods designed for functional data (where measurements are typically taken very frequently) to longitudinal data (with typically only a few measurements on each individual). However, these existing methods sometimes give fitted mean functions which are more complex than needed to provide a good fit to the data. I will describe a new penalised likelihood approach to flexibly model longitudinal data, with a penalty term to control the balance between fit to the data and smoothness of the subject-specific mean curves. I will show that the new method substantially improves the quality of inference relative to existing methods across a range of simulated examples, and apply the method to data on changes in body composition in adolescent girls.