  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 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, the
equivalence of statistical ensembles is violated and a clusterized phase is
expected which is not accessible in the grand-canonical ensemble. A specific
quasi-particle model will be introduced to illustrate this anomalous
thermodynamics 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 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.
