Iste magistrorum locus est simul et puerorum, mittunt quando volunt hic res quas perdere nolunt [1]. AbstractWe review the new possibilities offered by the reaction dynamics of asymmetric heavy ion collisions, using stable and unstable beams. We show that it represents a rather unique tool to probe regions of highly Asymmetric Nuclear Matter (AN M ) in compressed as well as dilute phases, and to test the in-medium isovector interaction for high momentum nucleons. The focus is on a detailed study of the symmetry term of the nuclear Equation of State (EOS) in regions far away from saturation conditions but always under laboratory controlled conditions. Thermodynamic properties of AN M are surveyed starting from nonrelativistic and relativistic effective interactions. In the relativistic case the role of the isovector scalar δ-meson is stressed. The qualitative new features of the liquid-gas phase transition, "diffusive" instability and isospin distillation, are discussed. The results of ab-initio simulations of n-rich, n-poor, heavy ion collisions, using stochastic isospin dependent transport equations, are analysed as a function of beam energy and centrality. The isospin dynamics plays an important role in all steps of the reaction, from prompt nucleon emissions to the final fragments. The isospin diffusion is also of large interest, due to the interplay of asymmetry and density gradients. In relativistic collisions, the possibility of a direct study of the covariant structure of the effective nucleon interaction is shown. Results are discussed for particle production, collective flows and iso-transparency.Perspectives of further developments of the field, in theory as well as in experiment, are presented.

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We try to single out some qualitative new effects of the coupling to the δ-isovector-scalar meson introduced in a minimal way in a phenomenological hadronic field theory. Results for the equation of state (EOS) and the phase diagram of asymmetric nuclear matter (AN M ) are discussed. We stress the consistency of the δ-coupling introduction in a relativistic approach.New contributions to the slope and curvature of the symmetry energy and the neutron-proton effective mass splitting appear particularly interesting.A more repulsive EOS for neutron matter at high baryon densities is expected.Effects on new critical properties of warm AN M , mixing of mechanical and chemical instabilities and isospin distillation, are also presented. The δ influence is mostly on the isovectorlike collective response.The results are largely analytical and this makes the physical meaning quite transparent. Implications for nuclear structure properties of drip-line nuclei and for reaction dynamics with Radioactive Beams are finally pointed out.