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The movements of individuals within and among cities influence critical aspects of our society, such as well-being, the spreading of epidemics, and the quality of the environment. When information about mobility flows is not available for a particular region of interest, we must rely on mathematical models to generate them. In this work, we propose the Deep Gravity model, an effective method to generate flow probabilities that exploits many variables (e.g., land use, road network, transport, food, health facilities) extracted from voluntary geographic data, and uses deep neural networks to discover non-linear relationships between those variables and mobility flows. Our experiments, conducted on mobility flows in England, Italy, and New York State, show that Deep Gravity has good geographic generalization capability, achieving a significant increase in performance (especially in densely populated regions of interest) with respect to the classic gravity model and models that do not use deep neural networks or geographic data. We also show how flows generated by Deep Gravity may be explained in terms of the geographic features using explainable AI techniques.