Problems of regression smoothing and curve fitting are addressed via predictive inference in a flexible class of mixture models. Multidimensional density estimation using Dirichlet mixture models provides the theoretical basis for semi-parametric regression methods in which fitted regression functions may be deduced as means of conditional predictive distributions. These Bayesian regression functions have features similar to generalised kernel regression estimates, but the formal analysis addresses problems of multivariate smoothing, parameter estimation, and the assessment of uncertainties about regression functions naturally. Computations are based on multidimensional versions of existing Markov chain simulation analysis of univariate Dirichlet mixture models.