  The effects of density dependence of the symmetry energy on the collective
modes and dynamical instabilities of cold and warm nuclear and stellar matter
are studied in the framework of relativistic mean-field hadron models. The
existence of the collective isovector and possibly an isoscalar collective mode
above saturation density is discussed. It is shown that soft equations of state
do not allow for a high density isoscalar collective mode, however, if the
symmetry energy is hard enough an isovector mode will not disappear at high
densities. The crust-core transition density and pressure are obtained as a
function of temperature for $\beta$-equilibrium matter with and without
neutrino trapping. An estimation of the size of the clusters formed in the
non-homogeneous phase as well as the corresponding growth rates and
distillation effect is made. It is shown that cluster sizes increase with
temperature, that the distillation effect close to the inner edge of the
crust-core transition is very sensitive to the symmetry energy, and that,
within a dynamical instability calculation, the pasta phase exists in warm
compact stars up to 10 - 12 MeV.
