Biological data may be correlated due to phylogenetic relationships, and may also be more naturally represented by continuous functions than by vectors. I will give a flexible statistical model for such data, by combining assumptions from phylogenetics with Gaussian processes. The model provides a prior distribution for Bayesian inference, enabling both prediction (for ancestral functions) and model selection. This work extends the popular phylogenetic Brownian Motion and Ornstein-Uhlenbeck models to functional data, and extends Gaussian Process regression to phylogenies. I will describe how Principal and Independent Components Analysis may be combined to provide a straightforward practical implementation of this phylogenetic regression for functional data.