This chapter examines the problem of binary classification under sample selection bias from a decision-theoretic perspective. Starting from a derivation of the necessary and sufficient conditions for equivalence of the Bayes classifiers of training and test distributions, it provides the conditions under which sample selection bias does not affect the performance of a classifier. From this viewpoint, there are fundamental differences between classifiers of low and high capacity, in particular the ones that are Bayes consistent. The second part of the chapter provides means to modify existing learning algorithms such that they are more robust to sample selection bias in the case where one has access to an unlabeled sample of the test data. This is achieved by constructing a graph-based regularization functional. The close connection of this approach to semisupervised learning is also highlighted.