In the analysis of spatial data, it is common to predict a spatial exceedance and its associated exceedance region. This is scientifically important, because unusual events tend to strongly affect the environment. We use classes of loss functions based on image metrics (e.g., Baddeley's loss function) to predict the spatial-exceedance region. We then propose a joint loss to predict a spatial quantile and its exceedance region. The optimal predictor is obtained by minimizing the posterior expected loss given the process parameters, which we achieve by simulated annealing. Various predictors are compared through simulation. This methodology is applied to a spatial data set of temperature change over the Americas. © 2008 American Statistical Association and the American Society for Quality.