Electroencephalography (EEG) is a complex signal and can require several years of training to be correctly interpreted. Recently, deep learning (DL) has shown great promise in helping make sense of EEG signals due to its capacity to learn good feature representations from raw data. Whether DL truly presents advantages as compared to more traditional EEG processing approaches, however, remains an open question. In this work, we review 156 papers that apply DL to EEG, published between January 2010 and July 2018, and spanning different application domains such as epilepsy, sleep, brain-computer interfacing, and cognitive and affective monitoring. We extract trends and highlight interesting approaches in order to inform future research and formulate recommendations. Various data items were extracted for each study pertaining to 1) the data, 2) the preprocessing methodology, 3) the DL design choices, 4) the results, and 5) the reproducibility of the experiments. Our analysis reveals that the amount of EEG data used across studies varies from less than ten minutes to thousands of hours. As for the model, 40% of the studies used convolutional neural networks (CNNs), while 14% used recurrent neural networks (RNNs), most often with a total of 3 to 10 layers. Moreover, almost one-half of the studies trained their models on raw or preprocessed EEG time series. Finally, the median gain in accuracy of DL approaches over traditional baselines was 5.4% across all relevant studies. More importantly, however, we noticed studies often suffer from poor reproducibility: a majority of papers would be hard or impossible to reproduce given the unavailability of their data and code. To help the field progress, we provide a list of recommendations for future studies and we make our summary table of DL and EEG papers available and invite the community to contribute.