  Heavy Ion Collisions (HIC) represent a unique tool to probe the in-medium
nuclear interaction in regions away from saturation. In this work we present a
selection of reaction observables in dissipative collisions particularly
sensitive to the isovector part of the interaction, i.e. to the symmetry term
of the nuclear Equation of State (EoS). At low energies the behavior of the
symmetry energy around saturation influences dissipation and fragment
production mechanisms. We will first discuss the recently observed Dynamical
Dipole Radiation, due to a collective neutron-proton oscillation during the
charge equilibration in fusion and deep-inelastic collisions. Important Iso-EOS
effects are stressed. Reactions induced by unstable 132Sn beams appear to be
very promising tools to test the sub-saturation Isovector EoS. New Isospin
sensitive observables are also presented for deep-inelastic, fragmentation
collisions and Isospin equilibration measurements (Imbalance Ratios). The high
density symmetry term can be derived from isospin effects on heavy ion
reactions at relativistic energies (few AGeV range), that can even allow a
``direct'' study of the covariant structure of the isovector interaction in the
hadron medium. Rather sensitive observables are proposed from collective flows
and from pion/kaon production. The possibility of the transition to a mixed
hadron-quark phase, at high baryon and isospin density, is finally suggested.
Some signatures could come from an expected ``neutron trapping'' effect. The
importance of studying violent collisions with radioactive beams from low to
relativistic energies is finally stressed.
