Recent advances in deep learning have led to a promising performance in many medical image analysis tasks. As the most commonly performed radiological exam, chest radiographs are a particularly important modality for which a variety of applications have been researched. The release of multiple, large, publicly available chest X-ray datasets in recent years has encouraged research interest and boosted the number of publications. In this paper, we review all studies using deep learning on chest radiographs published before March 2021, categorizing works by task: image-level prediction (classification and regression), segmentation, localization, image generation and domain adaptation. Detailed descriptions of all publicly available datasets are included and commercial systems in the field are described. A comprehensive discussion of the current state of the art is provided, including caveats on the use of public datasets, the requirements of clinically useful systems and gaps in the current literature.