In binary classification there are two types of errors, and in many applications these may have very different costs. We consider two learning frameworks that address this issue: minimax classification, where we seek to minimize the maximum of the false alarm and miss rates, and Neyman-Pearson (NP) classification, where we seek to minimize the miss rate while ensuring the false alarm rate is less than a specified level a. We show that our approach, based on cost-sensitive support vector machines, significantly outperforms methods typically used in practice. Our results also illustrate the importance of heuristics for improving the accuracy of error rate estimation in this setting. We then reduce anomaly detection to NP classification by considering a second class of points, allowing us to estimate minimum volume sets using algorithms for NP classification. Comparing this approach with traditional one-class methods, we find that our approach has several advantages.