There has been considerable recent interest in the nuclear equation of state and how it may be determined in relativistic nucleus-nucleus collisions. In these collisions extremely high temperatures are reached and compression to densities several times that of normal nuclear matter are predicted. This affords us the unique opportunity to study, in a somewhat controlled manner, the behavior of nuclear matter under these extreme conditions. If the observables that are measured in experiments can be related in a quantitative way to state variables of the system then the equation of state can be extracted. This relation plays a very important role in understanding the formation and collapse of supernovae and the stability and structure of neutron stars. Furthermore, it can be used to test and constrain field theoretical approaches to nuclear matter and to help to better understand the dynamics of high energy nucleus-nucleus collisions. In this presentation the relationship between the nuclear equation of state and relativistic nucleus-nucleus collisions will be discussed with an emphasis on how to extract the former. That a high density state of the collision should exist will be shown. One observable, namely the pion multiplicity, will be shown to survive the succeeding stages of the collision process to provide information on the equation of state at high densities. The resulting equation of state will be presented and discussed in the light of recent theoretical development. 34 refs., 12 figs.