  We 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 ($ANM$) 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 $ANM$ are surveyed starting from nonrelativistic
and relativistic effective interactions. In the relativistic case the role of
the isovector scalar $\delta$-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.
