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New technologies are transforming medicine, and this revolution starts with data. Health data, clinical images, genome sequences, data on prescribed therapies and results obtained, data that each of us has helped to create. Although the first uses of artificial intelligence (AI) in medicine date back to the 1980s, it is only with the beginning of the new millennium that there has been an explosion of interest in this sector worldwide. We are therefore witnessing the exponential growth of health-related information with the result that traditional analysis techniques are not suitable for satisfactorily management of this vast amount of data. AI applications (especially Deep Learning), on the other hand, are naturally predisposed to cope with this explosion of data, as they always work better as the amount of training data increases, a phase necessary to build the optimal neural network for a given clinical problem. This paper proposes a comprehensive and in-depth study of Deep Learning methodologies and applications in medicine. An in-depth analysis of the literature is presented; how, where and why Deep Learning models are applied in medicine are discussed and reviewed. Finally, current challenges and future research directions are outlined and analysed.