In nuclear collisions induced by stable or radioactive neutron-rich nuclei a transient state of nuclear matter with an appreciable isospin asymmetry as well as thermal and compressional excitation can be created. This offers the possibility to study the properties of nuclear matter in the region between symmetric nuclear matter and pure neutron matter. In this review, we discuss recent theoretical studies of the equation of state of isospin-asymmetric nuclear matter and its relations to the properties of neutron stars and radioactive nuclei. Chemical and mechanical instabilities as well as the liquid-gas phase transition in asymmetric nuclear matter are investigated. The in-medium nucleon-nucleon cross sections at different isospin states are reviewed as they affect significantly the dynamics of heavy ion collisions induced by radioactive beams. We then discuss an isospin-dependent transport model, which includes different mean-field potentials and cross sections for the proton and neutron, and its application to these reactions. Furthermore, we review the comparisons between theoretical predictions and available experimental data. In particular, we discuss the study of nuclear stopping in terms of isospin equilibration, the dependence of nuclear collective flow and balance energy on the isospin-dependent nuclear equation of state and cross sections, the isospin dependence of total nuclear reaction cross sections, and the role of isospin in preequilibrium nucleon emissions and subthreshold pion production.