We introduce a simple, yet powerful student-teacher framework for the challenging problem of unsupervised anomaly detection and pixel-precise anomaly segmentation in high-resolution images. To circumvent the need for prior data labeling, student networks are trained to regress the output of a descriptive teacher network that was pretrained on a large dataset of patches from natural images. Anomalies are detected when the student networks fail to generalize outside the manifold of anomaly-free training data, i.e., when the output of the student networks differ from that of the teacher network. Additionally, the intrinsic uncertainty in the student networks can be used as a scoring function that indicates anomalies. We compare our method to a large number of existing deep-learning-based methods for unsupervised anomaly detection. Our experiments demonstrate improvements over state-of-the-art methods on a number of real-world datasets, including the recently introduced MVTec Anomaly Detection dataset that was specifically designed to benchmark anomaly segmentation algorithms.