The equation of state was originally developed for ideal gases, and proved central to the development of early molecular and atomic physics. Increasingly sophisticated equations of state have been developed to take into account molecular interactions, quantization, relativistic effects and so on. Extreme conditions of matter are encountered both in nature and in the laboratory, for example in the centres of stars, in relativistic collisions of heavy nuclei, in inertial confinement fusion (where a temperature of 109 K and a pressure exceeding a billion atmospheres can be achieved). A sound knowledge of the equation of state is a prerequisite for understanding processes at very high temperatures and pressures, as noted in some recent developments. This book presents a detailed pedagogical account of the equation of state and its applications in several important and fast-growing topics in theoretical physics, chemistry and engineering. It should be of interest to upper-level undergraduates, graduate students, researchers and academics in theoretical physics, theoretical and nuclear chemistry, materials and mechanical engineering.