In this talk, we discuss the dissipative structure for a hyperbolic system with relaxation. If the relaxation term of the system has symmetric property, Shizuta-Kawashima(1984,1985) introduced a stability condition which induces the decay estimate for the solution of the Cauchy problem. However, there are some complicated physical models which possess a non-symmetric relaxation term and we can not apply this stability condition to these models. Under this situation, our purpose of this talk is to extend the stability condition for complicated models and get the quantitative decay estimate. Furthermore, we shall explain the new dissipative structure by using the several concrete examples.