We review current progress in the neutron scattering studies of magnetic excitations in high-transition temperature (high-Tc) copper oxide superconductors. Since its discovery 20 years ago, understanding the microscopic origin of superconductivity in these copper oxides has been the"holy-grill" in condensed matter physics. In contrast to conventional superconductors, where the interaction that pairs the electrons to form the superconducting state is mediated by lattice vibrations (phonons), it is generally believed that magnetic excitations might play a fundamental role in the superconducting mechanism of copper oxides because superconductivity occurs when mobile "electrons" or "holes" are doped into the antiferromagnetic parent compounds. In this review, we summarize the key results on the magnetic excitations obtained by inelastic neutron scattering over the past 20 years. We discuss the status of the field and point out possible future directions.