Median regression models become an attractive alternative to mean regression models when employing flexible families of distributions for the errors. Classical approaches are typically algorithmic with desirable properties emerging asymptotically. However, nonparametric error models may be most attractive in the case of smaller sample sizes where parametric specifications are difficult to justify. Hence, a Bayesian approach, enabling exact inference given the observed data, may be appealing. In this context there is little Bayesian work. We develop two fully Bayesian modeling approaches, employing mixture models, for the errors in a median regression model. The associated families of error distributions allow for increased variability, skewness, and flexible tail behavior. The first family is semiparametric with extra variability captured nonparametrically through mixing and skewness handled parametrically. The second family, a fully nonparametric one, includes all unimodal densities on the real line with median (and mode) equal to zero. Inconjunction with a parametric regression specification, two semiparametric median regression models arise. After fitting such models by using Gibbs sampling, full posterior inference for general population functionals is possible. The approach can also be applied when censored observations are present, leading to semiparametric censored median regression modeling. We illustrate with two examples, one involving censoring. © 2001, Taylor & Francis Group, LLC. All rights reserved.