The study of human mobility is crucial due to its impact on several aspects of our society, such as disease spreading, urban planning, well-being, pollution, and more. The proliferation of digital mobility data, such as phone records, GPS traces, and social media posts, combined with the outstanding predictive power of artificial intelligence, triggered the application of deep learning to human mobility. In particular, the literature is focusing on three tasks: next-location prediction, i.e., predicting an individual's future locations; crowd flow prediction, i.e., forecasting flows on a geographic region; and trajectory generation, i.e., generating realistic individual trajectories. 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: (i) basic notions on mobility and deep learning; (ii) a review of data sources and public datasets; (iii) a description of deep learning models and (iv) a discussion about relevant open challenges. Our survey is a guide to the leading deep learning solutions to next-location prediction, crowd flow prediction, and trajectory generation. At the same time, it helps deep learning scientists and practitioners understand the fundamental concepts and the open challenges of the study of human mobility.