The linear response of an equilibrium system to a perturbation by a conservative force is given by the well-known fluctuation-dissipation theorem. The theorem relates the linear response function to a correlation in the unperturbed dynamics of the observable with the entropy flux. In chapter 3 we study how this relation can be generalized to systems with non-equilibrium dynamics. We find that the linear response function can be written as a sum of correlation functions in the unperturbed non-equilibrium dynamics. One of the terms is again a correlation between the observable and the entropy flux due to the perturbation. The other term is a correlation between the observable and a new, less familiar quantity known as dynamical activity. We give a general algorithm for calculating the generalized fluctuation-dissipation relation and we apply it to a number of relevant examples.An important topic in thermodynamics is the study of the heat due totemperature changes, known as calorimetry. The study of heat capacities atlow temperatures was historically one of the motivations for the developmentof the quantum theory. But this study was limited to equilibrium systems. Inchapter 4 we study the Clausius heat theorem in a non-equilibrium setting,and we propose a possible extension of the concept of heat capacities for suchsystems. We show that unlike its equilibrium counterpart, this heat capacitycan be negative. We study this heat capacity by several worked-out examples.