This book grew out of an idea to study properties of small subsystems of a large reservoir. Observations were at the time not explainable with standard thermodynamics. But the theory of Hill on thermodynamics of small systems provided the systematic procedure needed to address the problem. Following Hill, thermodynamics can be formulated for the nanoscale! The purpose of this book is to expand and demonstrate Hill's theory. The theory adds a new term to the fundamental Gibbs equation, that is specific for systems at the nanoscale. The properties that follow may be counter intuitive. The equation of state for a small system, for instance, is not given once and for all. We shall see that it changes with the environmental variables that control the small system. The statistical mechanical machinery remains as before, however. The world of small systems challenges the standard knowledge; that the number of particles in a system must be very large for thermodynamic equations to apply. We shall see that thermodynamic equations apply perfectly well also for small particle numbers, provided that small-system effects are accounted for correctly. In the world where size and shape are central, we shall find that equations of state can be used down to one particle in a box! There are scaling laws, which help us determine and understand the large system limit better! In the first part, the authors highlight the basic idea of the theory and provide a more systematic method, than used before. In the second part, the authors demonstrate the power of the theory in a set of central applications of nanoscience in and away from equilibrium, for other scientists to be inspired for further use.