Many applications of AI involve scoring individuals using a learned function of their attributes. These predictive risk scores are then used to take decisions based on whether the score exceeds a certain threshold, which may vary depending on the context. The level of delegation granted to such systems in critical applications like credit lending and medical diagnosis will heavily depend on how questions of fairness can be answered. In this paper, we study fairness for the problem of learning scoring functions from binary labeled data, a classic learning task known as bipartite ranking. We argue that the functional nature of the ROC curve, the gold standard measure of ranking accuracy in this context, leads to several ways of formulating fairness constraints. We introduce general families of fairness definitions based on the AUC and on ROC curves, and show that our ROC-based constraints can be instantiated such that classifiers obtained by thresholding the scoring function satisfy classification fairness for a desired range of thresholds. We establish generalization bounds for scoring functions learned under such constraints, design practical learning algorithms and show the relevance our approach with numerical experiments on real and synthetic data.