The stability of a predator-prey model with time delay and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, sufficient conditions are given respectively for the local stability of each of feasible equilibria of the system and the existence of a Hopf bifurcation at the positive equilibrium. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the positive equilibrium exists. By using the Lyapunov functions and the LaSalle invariant principle, sufficient conditions are derived respectively for the global stability of each of feasible equilibria of the model.