We investigate the decay rates of correlations for nonuniformly hyperbolic systems with or without singularities, on piecewise H\"older observables. By constructing a new scheme of coupling methods using the probability renewal theory, we obtain the optimal bounds for decay rates of correlations for a large class of observables. Our results apply to rather general hyperbolic systems, including Bunimovich Stadia, Bunimovich billiards, semidispersing illiards on a rectangle and billiards with cusps, and to a wide class of nonuniformly hyperbolic maps. This is a joint work with Sandro Vaienti.