Since its genesis, quantum mechanics has proved to be a very accurate model for predicting the behavior of the world below the nanoscale. However, crucial breakthroughs in technology were needed in order to be able to e ectively access and manipulate such small magnitudes. During the last twenty years, the eld of quantum information processing has experienced a growing interest, in its many variants, both theoretically and practically. Despite being still at a very basic stage, expectations are high. The uniqueness of quantum phenomena (superposition of states, creation of entanglement, etc.) have no classical analogue and allow novelties such as another paradigm of computation, more secure com- munications, quantum teleportation, quantum dense coding, etc. which are presented and analyzed here. The aim of this Thesis is to present in a uni ed way the main mathematical methods used in quantum information processing, as well as the state of the art of their corresponding technological implementations. Our contributions are based in making a self-contained presentation; seeking completeness, that is,treating the most relevant elds of research involved, focusing on their relations; and picking the most relevant, insightful references, given the quantity of literature produced in this eld. We also discuss some of the fundamental questions that remain still unanswered, and which are the current lines of research to shed light on them.