Thermodynamics and statistical mechanics coexist within the larger theoretical edifice of thermal physics. And, particularly in undergraduate texts, it is heuristically advantageous to intermingle the macroscopic concepts of thermodynamics with the micro-picture provided by statistical mechanics. These observations notwithstanding, there are clearly some advantages – aesthetical, conceptual and mathematical – in producing an account of thermodynamics which makes no reference to the underlying microstructure of the system and that is one of our aims in this book. To this end we generalize the axiomaticalgebraic approach of Lieb and Yngvason to show that it yields the whole structure of thermodynamics, including the Carath´eodory version of the Second Law for both positive and negative temperatures and positive and negative heat capacities.We then compare this approach to the geometric approach of Boyling. In the concluding chapters we treat phase transitions and critical phenomena from a purely thermodynamic standpoint and describe the extension of the Lieb–Yngvason account to non-equilibrium, comparing this with the CIT model of non-equilibrium thermodynamics.