In this talk we introduce a mathematical framework -- Coupled Systems on Networks -- that describes a wide variety of physical, natural, and artificial complex dynamical systems: from biological and artificial neural networks, coupled systems of nonlinear oscillators on lattices, to complex ecosystems and the transmission models of infectious diseases in heterogeneous populations. We then give our new results (in collaboration with Michael Li) that address the global-stability problem for coupled systems on networks. Our results utilize a graph-theoretic approach to the constructions of Lyapunov functions for coupled systems on networks. We apply our results to several coupled systems in engineering, mathematical ecology, and mathematical epidemiology. Our results allow us to improve existing results in the literature and, particularly, resolve a 30-year open problem in mathematical epidemiology.