This book establishes the foundations of non-equilibrium quantum statistical mechanics in order to support students and academics in developing and building their understanding. The formal theory is derived from first principles by mathematical analysis, with concrete physical interpretations and worked examples throughout. It explains the central role of entropy; it's relation to the probability operator and the generalisation to transitions, as well as providing first principles derivation of the von Neumann trace form, the Maxwell-Boltzmann form and the Schr odinger equation.