A common problem in epidemiology, environmental science, econometrics, and other disciplines is to assess the relationships between response variables and covariates when both are measured over regions in two-dimensional space.  Examples include predicting county-by-county disease incidence or election outcomes using demographic, environmental, and/or economic variables.  A popular approach to accounting for spatial correlation in Bayesian models for such data is to include in the linear predictor a random effect for each region and to place a conditional-autoregressive (CAR) prior on the random effects.   For such models, this talk will survey possible motivations, implications for estimation of fixed effects, and various approaches to mitigating these effects.