This paper provides an overview of the theoretical solutions for shallow water equations and the characteristics of tropical waves. It also presents systematic methods for identifying equatorial waves from observations. Recent advancements in tropical wave observational studies have revealed several mechanisms that affect tropical wave evolutions, such as wave-flow interaction, wave-cumulus convection interaction, and moisture mode. Local large-scale circulation and cumulus convection play a critical role in controlling the structure, intensity, and movements of tropical waves. This paper also reviews the impact and mechanism of tropical waves on the formations and tracks of tropical cyclones. By examining tropical waves, the multiscale variation of tropical cyclone activity can be explained well. However, further research is necessary to address scientific problems related to atmospheric tropical waves to advance tropical atmospheric dynamics.