  Dense matter as it can be found in core-collapse supernovae and neutron stars
is expected to exhibit different phase transitions which impact the matter
composition and the equation of state, with important consequences on the
dynamics of core-collapse supernova explosion and on the structure of neutron
stars. In this paper we will address the specific phenomenology of two of such
transitions, namely the crust-core solid-liquid transition at sub-saturation
density, and the possible strange transition at super-saturation density in the
presence of hyperonic degrees of freedom. Concerning the neutron star
crust-core phase transition at zero and finite temperature, it will be shown
that, as a consequence of the presence of long-range Coulomb interactions, a
clusterized phase is expected which is not accessible in the grand-canonical
ensemble. A specific quasi-particle model will be introduced and some
quantitative results relevant for the supernova dynamics will be shown. The
opening of hyperonic degrees of freedom at higher densities corresponding to
the neutron stars core also modifies the equation of state. The general
characteristics and order of phase transitions in this regime will be analyzed
in the framework of a self-consistent mean-field approach.
