Variable selection and Bayesian effect fusion for categorical predictors in linear and logistic regression models. Effect fusion aims at the question which categories have a similar effect on the response and therefore can be fused to obtain a sparser representation of the model. Effect fusion and variable selection can be obtained either with a prior that has an interpretation as spike and slab prior on the level effect differences or with a sparse finite mixture prior on the level effects. The regression coefficients are estimated with a flat uninformative prior after model selection or by taking model averages. Posterior inference is accomplished by an MCMC sampling scheme which makes use of a data augmentation strategy (Polson, Scott & Windle (2013)) based on latent Polya-Gamma random variables in the case of logistic regression. The code for data augmentation is taken from Polson et al. (2013), who own the copyright.