The aim of this talk is to study the asymptotic behavior of solutions for some reaction-diffusion systems. By an entire solution, we mean a solution of the reaction-diffusion system which is guaranteed to exist for all real-time in the whole Euclidean space. We establish a Liouville-type theorem for the entire solutions of reaction-diffusion systems. Based on this theorem, we derive the stabilization of the solutions of the reaction-diffusion system to the unique positive constant state, under the condition that this positive constant state is globally stable in the corresponding kinetic systems. Several examples of spreading phenomena from ecology and epidemiology are given to illustrate the applications of this theory. Moreover, our theorem is generalized to some non-local diffusion problems. This is the joint work with Jong-Shenq Guo(Tamkang University) and Amy Poh Ai-Ling(University of Tokyo).