Research in quantum computation is looking for the consequences of having information encoding, processing and communication exploit the laws of quantum physics, i.e. the laws of the ultimate knowledge that we have, today, of the foreign world of elementary particles, as described by quantum mechanics. After an introduction to the principles of quantum information processing and a brief survey of the major breakthroughs brought by the first ten years of research in this domain, this paper concentrates on a typically “computer science” way to reach a deeper understanding of what it means to compute with quantum resources, namely on the design of programming languages for quantum algorithms and protocols, and on the questions raised by the semantics of such languages. Special attention is devoted to the process algebraic approach to such languages, through a presentation of QPAlg, the Quantum Process Algebra which is being designed by the authors.