Mammography is the most widely used method to screen breast cancer. Because of its mostly manual nature, variability in mass appearance, and low signal-to-noise ratio, a significant number of breast masses are missed or misdiagnosed. In this work, we present how Convolutional Neural Networks can be used to directly classify pre-segmented breast masses in mammograms as benign or malignant, using a combination of transfer learning, careful pre-processing and data augmentation to overcome limited training data. We achieve state-of-the-art results on the DDSM dataset, surpassing human performance, and show interpretability of our model.