We consider a Bayesian analysis of linear regression models that can account for skewed error distributions with fat tails.The latter two features are often observed characteristics of empirical data sets, and we will formally incorporate them in the inferential process.A general procedure for introducing skewness into symmetric distributions is first proposed.Even though this allows for a great deal of flexibility in distributional shape, tail behaviour is not affected.In addition, the impact on the existence of posterior moments in a regression model with unknown scale under commonly used improper priors is quite limited.Applying this skewness procedure to a Student-$t$ distribution, we generate a ``skewed Student'' distribution, which displays both flexible tails and possible skewness, each entirely controlled by a separate scalar parameter. The linear regression model with a skewed Student error term is the main focus of the paper: we first characterize existence of the posterior distribution and its moments, using standard improper priors and allowing for inference on skewness and tail parameters.For posterior inference with this model, a numerical procedure is suggested, using Gibbs sampling with data augmentation. The latter proves very easy to implement and renders the analysis of quite challenging problems a practical possibility.Two examples illustrate the use of this model in empirical data analysis.