The nuclear equation of state describes the relationship among the macroscopic quantities including the binding energy, pressure, density and the density difference of neutrons and protons in bulk nuclear matter. It is not only relevant to the nuclear force properties, nuclear structure and dynamics of heavy ion reactions, but also related to the thermodynamic properties, structure, evolution and merging events of dense stellar objects like neutron stars. As a unique terrestrial way to produce nuclear matter in extreme conditions far away from saturation point, heavy ion reactions based on particle accelerators provide an effective tool to study the nuclear equation of state in a ground laboratory. When the neutron density is much higher than the proton density, as is the situation in an inner neutron star, the main contribution in the nuclear equation of state comes from the symmetry energy term, which is a function of density. This dependence is the most important yet so far unknown quantity in nuclear physics and astrophysics. Thus, it is now a foremost challenge of modern nuclear physics to determine, through experimental and theoretical research on the heavy ion reactions, the dependence of the symmetry energy on density as well as its effects on nuclear reactions and astrophysics events involving dense stellar objects. This paper presents the background and methods in the study of medium energy heavy ion reactions and the nuclear equations of state, followed by a brief introduction to some research progress in recent years.