Abstract: This paper is concerned with a three-species delayed reaction-diffusion predator-prey system in a bounded domain with Neumann boundary condition. The sufficient conditions of stability are found for equilibria of this system by the method of eigenvalue and Lyapunov function, and these conditions imply that delays often restrain stability. One of the main results about stabilities shows that if the intra-specific competitions of the predator and preys dominate their inter-specific interaction, then the positive equilibria are globally stable. Furthermore, the existence of the traveling wavefront is considered by constructing upper-lower solution and it is derived that this system always has a traveling wave solution connecting the trivial steady state and the positive steady state when the wave speed is relatively big.

Key words: stability, delay, traveling waves