The monograph presents a comparative analysis of different thermodynamic models of the equations of state. The basic ideological premises of the theoretical methods and the experiment are considered. The principal attention is on the description of states that are of greatest interest for the physics of high energy concentrations which are either already attained or can be reached in the near future in controlled terrestrial conditions, or are realized in astrophysical objects at different stages of their evolution. Ultra-extreme astrophysical and nuclear-physical applications are also analyzed where the thermodynamics of matter is affected substantially by relativism, high-power gravitational and magnetic fields, thermal radiation, transformation of nuclear particles, nucleon neutronization, and quark deconfinement. The book is intended for a wide range of specialists engaged in the study of the equations of state of matter and high energy density physics, as well as for senior students and postgraduates.