Magnonics, an emerging research field, aims to control and manipulate spin waves in magnetic materials and structures. However, the current understanding of spin waves remains quite limited. This review attempts to provide an overview of the anomalous behaviors of spin waves in various types of magnetic materials observed thus far by inelastic light scattering experiments. The anomalously large asymmetry of anti-Stokes to Stokes intensity ratio, broad linewidth, strong resonance effect, unique polarization selection, and abnormal impurity dependence of spin waves are discussed. In addition, the mechanisms of these anomalous behaviors of spin waves are proposed. spectroscopy is applied for studying vibrational properties of materials. However, due to the anomalous behaviors of spin wave scattering comparing with vibrational scattering, Raman spectroscopy study of spin wave is difficult and limited. In this paper, we present a review of anomalous behaviors of spin waves observed by inelastic light scattering experiments, especially by Raman spectroscopy. The large asymmetry of anti-Stokes to Stokes intensity ratio, broad linewidth, strong resonance effect, unique polarization selection, and abnormal impurity dependence of spin waves are discussed, and a proposed model for the mechanisms of spin flip, spin relaxing, and spin wave scattering is presented for understanding these anomalous behaviors of spin waves. In addition, two magnonic crystal structures are proposed for manipulating spin waves through analyses of the abnormal impurity dependence of spin waves. The review of the anomalous behaviors of spin waves and the proposed models provide the directions for easier experimental study of spin waves by inelastic light scattering, which will be useful for raising the research efforts for investigating spin waves. This is important for the emerging research of magnonics, which has received extensive interests in the modern magnetism research community.