High-transition temperature (high-Tc) superconductivity in the iron pnictides or chalcogenides emerges from the suppression of the static antiferromagnetic order in their parent compounds, similar to copper oxide superconductors. This raises a fundamental question concerning the role of magnetism in the superconductivity of these materials. Neutron scattering, a powerful probe to study the magnetic order and spin dynamics, plays an essential role in determining the relationship between magnetism and superconductivity in high-Tc superconductors. The rapid development of modern neutron time-of-flight spectrometers allows a direct determination of the spin dynamical properties of iron-based superconductors throughout the entire Brillouin zone. In this paper, an overview is presented of the neutron scattering results on iron-based superconductors, focusing on the evolution of spin-excitation spectra as a function of electron and hole doping and isoelectronic substitution. Spin dynamical properties of iron-based superconductors are compared with those of copper oxide and heavy fermion superconductors and the common features of spin excitations in these three families of unconventional superconductors and their relationship with superconductivity are discussed.