Bayesian approaches to density estimation and clustering using mixture distributions allow the automatic determination of the number of components in the mixture. Previous treatments have focused on mixtures having Gaussian components, but these are well known to be sensitive to outliers. This can lead to excessive sensitivity to small numbers of data points and consequent over-estimates of the number of components. In this paper we develop a Bayesian approach to mixture modelling based on Student-t distributions, which are heavier tailed than Gaussians and hence more robust. By expressing the Student-t distribution as a marginalization over additional latent variables we are able to derive a tractable variational inference algorithm for this model, which includes Gaussian mixtures as a special case. Results on a variety of real data sets demonstrate the improved robustness of our approach.