The topological interface states in elastic periodic structures have attracted wide attention and research interest due to their unique transmission characteristics and eigenmodes with interface energy enhancement effects. This characteristic provides new ideas for designing novel elastic waveguides and developing quantum devices in next-generation information technology. Moreover, it is also valuable for classical wave modulation. This article systematically introduces the research progress of one-dimensional topological interface states in phonon and elastic wave systems, which elaborates on concepts related to topological research, such as band inversion, Zak phase with taking the Su-Schrieffer-Heeger (SSH) model as an example. It deeply explores one-dimensional topological interface states’ design ideas and current research progress. It also summarizes the active regulation methods of the eigenfrequency of topological states: multi-field coupling modulation and mechanically modulation. Finally, it explores possible directions for future research on topological interface states.