Quantum information science is one of the most dynamic and exciting areas of science today, its significance ranging from the ultimate physical limits of information processing, to fundamental issues of quantum mechanics, to new mathematics and prospects of realising novel, enhanced computation and communicationtechnologies.Drawing on ideas from physics, mathematics and computer science, one of the core programmes of this highly interdisciplinary field is to understand and harness quantum mechanics in terms of information theory.Past steps in this programme include Shor's quantum factoring algorithm, quantum cryptography, and the creation of quantum Shannon and entanglement theory. Dramatic recent developments, while opening up exciting new avenues, at the same time pose major challenges: the proof that quantum capacities can be non-additive showed that the landscape of quantum Shannon theory is actually much richer than previously envisaged; the information theory of generalized non-locality pointed out a completely new take on the foundations of quantum mechanics; and mathematical tools such as the probabilistic method have had a major impact across the board from entanglement theory to quantum Shannon theory and statistical mechanics.In the proposed research I will address key current open questions in quantum Shannon theory, including long-standing coding problems, and a deeper understanding of the pervasive non-additivity of quantum capacities; develop probabilistic tools directly related to quantum information; find new entropy inequalities and entropic uncertainty relations; characterise entanglement and non-local correlations in information theoretic terms; and realise an ambitious programme of founding statistical mechanics on generic quantum dynamics.