In the last few years, there has been a growing expectation created about the analysis of large amounts of data often available in organizations, which has been both scrutinized by the academic world and successfully exploited by industry. Nowadays, two of the most common terms heard in scientific circles are Big Data and Deep Learning. In this double review, we aim to shed some light on the current state of these different, yet somehow related branches of Data Science, in order to understand the current state and future evolution within the healthcare area. We start by giving a simple description of the technical elements of Big Data technologies, as well as an overview of the elements of Deep Learning techniques, according to their usual description in scientific literature. Then, we pay attention to the application fields that can be said to have delivered relevant real-world success stories, with emphasis on examples from large technology companies and financial institutions, among others. The academic effort that has been put into bringing these technologies to the healthcare sector are then summarized and analyzed from a twofold view as follows: first, the landscape of application examples is globally scrutinized according to the varying nature of medical data, including the data forms in electronic health recordings, medical time signals, and medical images; second, a specific application field is given special attention, in particular the electrocardiographic signal analysis, where a number of works have been published in the last two years. A set of toy application examples are provided with the publicly-available MIMIC dataset, aiming to help the beginners start with some principled, basic, and structured material and available code. Critical discussion is provided for current and forthcoming challenges on the use of both sets of techniques in our future healthcare.