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The proliferation of digital mobility data, such as phone records, GPS traces, and social media posts, combined with the predictive power of artificial intelligence, triggered the application of deep learning to human mobility. Existing surveys focus on single tasks, data sources, mechanistic or traditional machine learning approaches, while a comprehensive description of deep learning solutions is missing. This survey provides a taxonomy of mobility tasks, a discussion on the challenges related to each task and how deep learning may overcome the limitations of traditional models, a description of the most relevant solutions to the mobility tasks described above, and the relevant challenges for the future. Our survey is a guide to the leading deep learning solutions to next-location prediction, crowd flow prediction, trajectory generation, and flow generation. 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