This course provides an overview of the latest advancements in quantum information theory and quantum computation. Starting from the foundation of the framework of density matrices and open system dynamics, the course gradually transitions into two main chapters, the first about the quantum theory of information compression and transmission and the second on the topological models of quantum computation. The first chapter will deal with quantum entropies and quantum typicality, providing useful tools also for students interested in quantum thermodynamics. The second chapter will deal with anyons and geometric phases, showing the features of anyonic statistics can be used to perform stable and efficient quantum computations.Students who take this course will develop the mental discipline needed to identify and discuss critically these questions and will be provided with the sharpest theoretical tools to address these questions.