In recent years several techniques have been proposed for modelling the low-dimensional manifolds, or `subspaces’, of natural images. Examples include principal component analysis (as used for instance in `eigen-faces’), independent component analysis, and auto-encoder neural networks. Such methods suffer from a number of restrictions such as the limitation to linear manifolds or the absence of a probabilistic representation. In this paper we exploit recent developments in the fields of variational inference and latent variable models to develop a novel and tractable probabilistic approach to modelling manifolds which can handle complex non-linearities. Our framework comprises a mixture of sub-space components in which both the number of components and the effective dimensionality of the sub-spaces are determined automatically as part of the Bayesian inference procedure. We illustrate our approach using two classical problems: modelling the manifold of face images and modelling the manifolds of hand-written digits.