The past decade has seen an explosion in the amount of digital information stored in electronic health records (EHR). While primarily designed for archiving patient information and performing administrative healthcare tasks like billing, many researchers have found secondary use of these records for various clinical informatics applications. Over the same period, the machine learning community has seen widespread advances in the field of deep learning. In this review, we survey the current research on applying deep learning to clinical tasks based on EHR data, where we find a variety of deep learning techniques and frameworks being applied to several types of clinical applications including information extraction, representation learning, outcome prediction, phenotyping, and de-identification. We identify several limitations of current research involving topics such as model interpretability, data heterogeneity, and lack of universal benchmarks. We conclude by summarizing the state of the field and identifying avenues of future deep EHR research.