Transport theories for heavy ion collisions and the study of the nuclear EOS

Heavy ion collisions offer the possibility to explore, in laboratory experiments, transient states of nuclear matter under several conditions of density, temperature and charge asymmetry. Transport models are an essential tool to describe the reaction dynamics and make a connection between the nuclear effective interaction employed in the calculations and sensitive observables of experimental interest. We review here recent analyses which allow to constrain, in particular, the controversial density behavior of the symmetry energy of the nuclear EOS, also discussing future perspectives. (author)