We consider the problem of learning to simplify medical texts. This is important because most reliable, up-to-date information in biomedicine is dense with jargon and thus practically inaccessible to the lay audience. Furthermore, manual simplification does not scale to the rapidly growing body of biomedical literature, motivating the need for automated approaches. Unfortunately, there are no large-scale resources available for this task. In this work we introduce a new corpus of parallel texts in English comprising technical and lay summaries of all published evidence pertaining to different clinical topics. We then propose a new metric based on likelihood scores from a masked language model pretrained on scientific texts. We show that this automated measure better differentiates between technical and lay summaries than existing heuristics. We introduce and evaluate baseline encoder-decoder Transformer models for simplification and propose a novel augmentation to these in which we explicitly penalize the decoder for producing “jargon” terms; we find that this yields improvements over baselines in terms of readability.