Summary: We develop a flexible model-based procedure for clustering functional data. The technique can be applied to all types of curve data but is particularly useful when individuals are observed at a sparse set of time points. In addition to producing final cluster assignments, the procedure generates predictions and confidence intervals for missing portions of curves. Our approach also provides many useful tools for evaluating the resulting models. Clustering can be assessed visually via low-dimensional representations of the curves, and the regions of greatest separation between clusters can be determined using a discriminant function. Finally, we extend the model to handle multiple functional and finite-dimensional covariates and show how it can be applied to standard finite-dimensional clustering problems involving missing data.