Quantum thermodynamics is a newly emerging field of modern research, which has recently attracted more and more activity. Although classical thermodynamics and quantum mechanics were developed side by side in the beginning of the twentieth century the thermodynamic description of quantum systems has only recently become a major topic of modern research. This development has been spurred by the discovery of novel and fundamental generalizations of the second law of thermodynamics explicitly accounting for non-equilibrium fluctuations. These so-called fluctuation theorems changed the conventional focus of thermodynamics on work and heat to the study of entropy production — a concept more accessible in quantum systems. To this end, three main avenues of research have been pursued: identifying fundamental notions of entropy and information production in quantum systems, studying the thermodynamics of information processing, and analyzing optimal quantum processes. As a theoretical physicists, we employ tools from Statistical Physics, Open Quantum Dynamics, Quantum Information Theory, Quantum Optics, Condensed Matter Theory and Optimal Control Theory to investigate the nonequilibrium properties of nanosystems operating far from thermal equilibrium.