Deep learning (DL) is one of the branches of artificial intelligence that has seen exponential growth in recent years. The scientific community has focused its attention on DL due to its versatility, high performance, high generalization capacity, and multidisciplinary uses, among many other qualities. In addition, a large amount of medical data and the development of more powerful computers has also fostered an interest in this area. This paper presents an overview of current deep learning methods, starting from the most straightforward concept but accompanied by the mathematical models that are behind the functionality of this type of intelligence. In the first instance, the fundamental concept of artificial neural networks is introduced, progressively covering convolutional structures, recurrent networks, attention models, up to the current structure known as the Transformer. Secondly, all the basic concepts involved in training and other common elements in the design of the architectures are introduced. Thirdly, some of the key elements in modern networks for medical image classification and segmentation are shown. Subsequently, a review of some applications realized in the last years is shown, where the main features related to DL are highlighted. Finally, the perspectives and future expectations of deep learning are presented.