In this paper we consider regression models for count data allowing for overdispersion in a Bayesian framework. Besides the inclusion of covariates, spatial effects are incorporated and modelled using a proper Gaussian conditional autoregressive prior based on Pettitt et al. (2002). Apart from the Poisson regression model, the negative binomial and the generalized Poisson regression model are addressed. Further, zero-inflated models combined with the Poisson and generalized Poisson distribution are discussed. In an application to a data set from a German car insurance company we use the presented models to analyse the expected number of claims. Models are compared according to the deviance information criterion (DIC) suggested by Spiegelhalter et al. (2002). To assess the model fit we use posterior predictive p-values proposed by Gelman et al. (1996). For this data set no significant spatial effects are observed, however the models allowing for overdispersion perform better than a simple Poisson regression model.