Recent years have seen tremendous progress towards building the first small scale quantum computers and quantum communication networks. Companies such as Google, IBM and Microsoft have identified these fields as focus of their investment into the future. With these developments expected to continue, it will soon become crucial to better understand the possibilities and limitations of quantum information processing in order to make good use of the provided hardware. A significant part of these efforts will have to go into quantum information theory, following the lead of traditional information theory which currently provides the main source for advances in information transmission, such as new codes for the next generation of wireless networks (5G and WIFI6). At the heart of classical and quantum information theory is the study of entropic quantities, as they provide us with a tool to quantify the amount of information present in a system. In particular, entropy inequalities allow us to investigate what happens when we process information and to relate different operational tasks. Their use has already let to security proofs in classical and quantum cryptography, new codes for near optimal information transmission and much more. Moreover, the study of entropic quantities provides us with a fundamental understanding of information, that will aid the development of future applications. In order to facilitate the understanding of these topics, this Masterclass will gather some of the top researchers in the field to communicate recent developments and open problems that still need to be overcome.