The study of isospin physics via heavy-ion reactions with neutron-rich, stable, and radioactive nuclei aims to explore the isospin dependence of in-medium nuclear effective interactions and the equation of state (EOS) of neutron-rich nuclear matter, particularly the density dependence of the nuclear symmetry energy. This has been a major focus in intermediate-energy heavy-ion physics over the last decade, with significant experimental and theoretical progress. Key observables, such as neutron-proton ratios and flow patterns, have been identified as sensitive probes of the symmetry energy's density dependence. Experimental studies have constrained the symmetry energy at sub-saturation densities, and its impact on neutron star properties has been studied, showing its importance for astrophysics. New radioactive beam facilities are enabling further exploration of isospin physics, which remains a forefront area in nuclear physics. The EOS of isospin-asymmetric nuclear matter is a long-standing problem in nuclear and astrophysics. Theoretical approaches include microscopic many-body methods, effective-field theories, and phenomenological models. The microscopic many-body approach, such as the Brueckner-Hartree-Fock (BHF) and self-consistent Green's function (SCGF) methods, provides insights into the EOS, while effective-field theories, like density functional theory and chiral perturbation theory, offer systematic expansions of the EOS in terms of density. Phenomenological models, such as the relativistic mean-field (RMF) model, use effective interactions to describe nuclear properties and have been successful in explaining various nuclear phenomena. The RMF model, based on effective interaction Lagrangians, has been extended to include in-medium hadron properties and chiral symmetry restoration. It has been used to describe nuclear matter and finite nuclei, with parameter sets that match experimental data. The non-relativistic Hartree-Fock approach with Skyrme forces and the Thomas-Fermi approximation are also important tools in nuclear structure studies. These models help constrain the symmetry energy and its density dependence, which is crucial for understanding neutron stars and other astrophysical phenomena. Recent studies have shown that the symmetry energy at sub-saturation densities is well-constrained, while at higher densities, experimental data remains limited. Future research with new facilities will further advance our understanding of isospin physics and the EOS of neutron-rich nuclear matter.The study of isospin physics via heavy-ion reactions with neutron-rich, stable, and radioactive nuclei aims to explore the isospin dependence of in-medium nuclear effective interactions and the equation of state (EOS) of neutron-rich nuclear matter, particularly the density dependence of the nuclear symmetry energy. This has been a major focus in intermediate-energy heavy-ion physics over the last decade, with significant experimental and theoretical progress. Key observables, such as neutron-proton ratios and flow patterns, have been identified as sensitive probes of the symmetry energy's density dependence. Experimental studies have constrained the symmetry energy at sub-saturation densities, and its impact on neutron star properties has been studied, showing its importance for astrophysics. New radioactive beam facilities are enabling further exploration of isospin physics, which remains a forefront area in nuclear physics. The EOS of isospin-asymmetric nuclear matter is a long-standing problem in nuclear and astrophysics. Theoretical approaches include microscopic many-body methods, effective-field theories, and phenomenological models. The microscopic many-body approach, such as the Brueckner-Hartree-Fock (BHF) and self-consistent Green's function (SCGF) methods, provides insights into the EOS, while effective-field theories, like density functional theory and chiral perturbation theory, offer systematic expansions of the EOS in terms of density. Phenomenological models, such as the relativistic mean-field (RMF) model, use effective interactions to describe nuclear properties and have been successful in explaining various nuclear phenomena. The RMF model, based on effective interaction Lagrangians, has been extended to include in-medium hadron properties and chiral symmetry restoration. It has been used to describe nuclear matter and finite nuclei, with parameter sets that match experimental data. The non-relativistic Hartree-Fock approach with Skyrme forces and the Thomas-Fermi approximation are also important tools in nuclear structure studies. These models help constrain the symmetry energy and its density dependence, which is crucial for understanding neutron stars and other astrophysical phenomena. Recent studies have shown that the symmetry energy at sub-saturation densities is well-constrained, while at higher densities, experimental data remains limited. Future research with new facilities will further advance our understanding of isospin physics and the EOS of neutron-rich nuclear matter.