We introduce a powerful student-teacher framework for the challenging problem of unsupervised anomaly detection and pixel-precise anomaly segmentation in high-resolution images. 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. This circumvents the need for prior data annotation. Anomalies are detected when the outputs of the student networks differ from that of the teacher network. This happens when they fail to generalize outside the manifold of anomaly-free training data. The intrinsic uncertainty in the student networks is used as an additional 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.