The article reviews recent progress and new challenges in isospin physics with heavy-ion reactions, focusing on the exploration of the isospin dependence of in-medium nuclear effective interactions and the equation of state (EOS) of neutron-rich nuclear matter. The EOS of isospin-asymmetric nuclear matter is a key area of interest, particularly the density dependence of the nuclear symmetry energy. Significant progress has been made in both experimental and theoretical studies, leading to a better understanding of the properties of neutron stars and other astrophysical phenomena. The review covers various theoretical approaches, including microscopic many-body methods, effective-field theory, and phenomenological models, as well as experimental techniques for studying isospin effects in heavy-ion reactions. The article highlights the importance of next-generation radioactive beam facilities in advancing the field and discusses future challenges, such as the high-density behavior of the symmetry energy and the liquid-gas phase transition in asymmetric nuclear matter.The article reviews recent progress and new challenges in isospin physics with heavy-ion reactions, focusing on the exploration of the isospin dependence of in-medium nuclear effective interactions and the equation of state (EOS) of neutron-rich nuclear matter. The EOS of isospin-asymmetric nuclear matter is a key area of interest, particularly the density dependence of the nuclear symmetry energy. Significant progress has been made in both experimental and theoretical studies, leading to a better understanding of the properties of neutron stars and other astrophysical phenomena. The review covers various theoretical approaches, including microscopic many-body methods, effective-field theory, and phenomenological models, as well as experimental techniques for studying isospin effects in heavy-ion reactions. The article highlights the importance of next-generation radioactive beam facilities in advancing the field and discusses future challenges, such as the high-density behavior of the symmetry energy and the liquid-gas phase transition in asymmetric nuclear matter.