In this talk, we discuss recent advances in Bayesian logistic and linear regression models. Gelman et al. (2008) recommended Cauchy prior distributions for the regression coefficients in logistic regression models. As the mean does not exist for the Cauchy distribution, a natural question is whether the posterior means of the regression coefficients exist. We provide some conditions for the existence of posterior means and discuss the implications of these results in practice, using simulated and real datasets. In the second part of the talk, we focus on methods for Bayesian model averaging in linear regression models with error distributions that are heavier tailed than the normal distribution. We use simulated and real datasets to demonstrate some of the advantages of these methods compared to their traditional normal error counterparts.