||We establish the existence and the stability of traveling wave solutions of a quasi-linear hyperbolic system with both relaxation and diffusion. The traveling wave solutions are shown to be asymptotically stable under small
disturbances and under the sub-characteristic condition using a weighted energy method. The delicate balance between the relaxation and the diffusion that leads to the stability of the traveling waves is identified, namely, the
diffusion coefficient is bounded by a constant multiple of the relaxation time. Such a result provides an important first step toward the understanding of the transition from stability to instability as parameters vary in physical
problems involving both relaxation and diffusion.