Abstract The application of hydrodynamics to the study of heavy ion collisions is presented with emphasis on the relativistic regime. Current theoretical and experimental knowledge of the nuclear equations of state along with the possible roles played by viscosity, single- and double-shock waves and solitons during the collision are examined. Constraints imposed by relativity on the equation of state and numerical results from one- and three-dimensional one-fluid relativistic calculations are reviewed. Also considered are sources of entropy production arising from irreversible processes. Some alterations and additions to the basic equations of relativistic hydrodynamics are proposed to describe phenomena associated with relativistic heavy ion collisions. The equations of relativistic chromohydrodynamics are reviewed as a possible framework in which to consider the nuclear to quark-gluon phase transition. Transparency is introduced by a two-fluid model; results for center-of-mass bombarding energies up to 20 GeV per nucleon are presented. Recent work on pion production, the entropy problem and global analysis are reviewed.