In this thesis we present novel methods for constructing and fitting 2d models of shape and appearance which are used for analysing human faces. The first contribution builds on previous work on discriminative fitting strategies for active appearance models (AAMs) in which regression models are trained to predict the location of shapes based on texture samples. In particular, we investigate non-parametric regression methods including random forests and Gaussian processes which are used together with gradient-like features for shape model fitting. We then develop two training algorithms which combine such models into sequences, and systematically compare their performance to existing linear generative AAM algorithms. Inspired by the performance of the Gaussian process-based regression methods, we investigate a group of non-linear latent variable models known as Gaussian process latent variable models (GPLVM). We discuss how such models may be used to develop a generative active appearance model algorithm whose texture model component is non-linear, and show how this leads to lower-dimensional models which are capable of generating more natural-looking images of faces when compared to equivalent linear models. We conclude by describing a novel supervised non-linear latent variable model based on Gaussian processes which we apply to the problem of recognising emotions from facial expressions.