Visual speech synthesis deals with synthesising facial animation from an audio representation of speech. In the last decade or so, data-driven approaches have gained prominence with the development of Machine Learning techniques that can learn an audio-visual mapping. Many of these Machine Learning approaches learn a generative model of speech production using the framework of probabilistic graphical models, through which efficient inference algorithms can be developed for synthesis.In this work, the audio and visual parameters are assumed to be generated from an underlying latent space that captures the shared information between the two modalities. These latent points evolve through time according to a dynamical mapping and there are mappings from the latent points to the audio and visual spaces respectively. The mappings are modelled using Gaussian processes, which are non-parametric models that can represent a distribution over non-linear functions. The result is a non-linear state-space model. It turns out that the state-space model is not a very accurate generative model of speech production because it assumes a single dynamical model, whereas it is well known that speech involves multiple dynamics (for e.g. different syllables) that are generally non-linear. In order to cater for this, the state-space model can be augmented with switching states to represent the multiple dynamics, thus giving a switching state-space model. A key problem is how to infer the switching states so as to model the multiple non-linear dynamics of speech, which we address by learning a variable-order Markov model on a discrete representation of audio speech. Various synthesis methods for predicting visual from audio speech are proposed for both the state-space and switching state-space models.Quantitative evaluation, involving the use of error and correlation metrics between ground truth and synthetic features, is used to evaluate our proposed method in comparison to other probabilistic models previously applied to the problem. Furthermore, qualitative evaluation with human participants has been conducted to evaluate the realism, perceptual characteristics and intelligibility of the synthesised animations. The results are encouraging and demonstrate that by having a joint probabilistic model of audio and visual speech that caters for the non-linearities in audio-visual mapping, realistic visual speech can be synthesised from audio speech.