The symmetry energy contribution to the nuclear equation of state impacts various phenomena in nuclear astrophysics, nuclear structure, and nuclear reactions. Its determination is a key objective of contemporary nuclear physics, with consequences for the understanding of dense matter within neutron stars. We examine the results of laboratory experiments that have provided initial constraints on the nuclear symmetry energy and on its density dependence at and somewhat below normal nuclear matter density. Even though some of these constraints have been derived from properties of nuclei while others have been derived from the nuclear response to electroweak and hadronic probes, within experimental uncertainties-they are consistent with each other. We also examine the most frequently used theoretical models that predict the symmetry energy and its slope parameter. By comparing existing constraints on the symmetry pressure to theories, we demonstrate how contributions of three-body forces, which are essential ingredients in neutron matter models, can be determined.

We perform a systematic analysis of the density dependence of nuclear symmetry energy within the microscopic Brueckner-Hartree-Fock (BHF) approach using the realistic Argonne V18 nucleon-nucleon potential plus a phenomenological three-body force of Urbana type. Our results are compared thoroughly with those arising from several Skyrme and relativistic effective models. The values of the parameters characterizing the BHF equation of state of isospin asymmetric nuclear matter fall within the trends predicted by those models and are compatible with recent constraints coming from heavy ion collisions, giant monopole resonances, or isobaric analog states. In particular we find a value of the slope parameter $L=66.5$ MeV, compatible with recent experimental constraints from isospin diffusion, $L=88\pm 25$ MeV. The correlation between the neutron skin thickness of neutron-rich isotopes and the slope $L$ and curvature $K_{\mathrm{sym}}$ parameters of the symmetry energy is studied. Our BHF results are in very good agreement with the correlations already predicted by other authors using nonrelativistic and relativistic effective models. The correlations of these two parameters and the neutron skin thickness with the transition density from nonuniform to $\beta$-stable matter in neutron stars are also analyzed. Our results confirm that there is an inverse correlation between the neutron skin thickness and the transition density.

We study the consequences of the hadron-quark deconfinement phase transition in stellar compact objects when finite size effects between the deconfined quark phase and the hadronic phase are taken into account. We show that above a threshold value of the central pressure (gravitational mass) a neutron star is metastable to the decay (conversion) to a hybrid neutron star or to a strange star. The "mean-life time" of the metastable configuration dramatically depends on the value of the stellar central pressure. We explore the consequences of the metastability of ``massive'' neutron stars and of the existence of stable compact quark stars (hybrid neutron stars or strange stars) on the concept of limiting mass of compact stars. We discuss the implications of our scenario on the interpretation of the stellar mass and radius extracted from the spectra of several X-ray compact sources. Finally, we show that our scenario implies, as a natural consequence a two step-process which is able to explain the inferred ``delayed'' connection between supernova explosions and GRBs, giving also the correct energy to power GRBs

The possibility to draw links between the isospin properties of nuclei and the structure of compact stars is a stimulating perspective. In order to pursue this objective on a sound basis, the correlations from which such links can be deduced have to be carefully checked against model dependence. Using a variety of nuclear effective models and a microscopic approach, we study the relation between the predictions of a given model and those of a Taylor density development of the corresponding equation of state: this establishes to what extent a limited set of phenomenological constraints can determine the core-crust transition properties. From a correlation analysis, we show that (a) the transition density $\rho${}t is mainly correlated with the symmetry energy slope L, (b) the proton fraction Yp,t with the symmetry energy and symmetry energy slope (J,L) defined at saturation density, or, even better, with the same quantities defined at $\rho${}=0.1 fm-3, and (c) the transition pressure Pt with the symmetry energy slope and curvature (L,Ksym) defined at $\rho${}=0.1 fm-3.