This volume is published in honor of Friedrich Hund's 100th birthday. It is a modern review on matter at high densities and pressures in astrophysics from Hund's early contribution to present-day ideas. The relation between the equation of state and the structure of compact cosmic objects is discussed, and two main contributions deal with the equation of state of baryonic matter at nuclear densities and with the numerical solution of the general relativistic field equations for non-rotating and rapidly rotating neutron stars. In a final chapter the present state of asteroseismology is presented as a tool to explore the interior of cosmic objects by analyzing the observed free oscillations of the Earth, the Sun, and white dwarf stars.