In this thesis, we study stability and stabilizability problems in the framework of coupled dynamic systems. Particular attention is given to the class of coupled dynamic systems whose equilibrium set is described by all states having identical state components. Central to the stability and stabilizability issues of such systems is the graph describing the interaction structure---that is, who is coupled to whom. A central question is, what properties of the interaction graphs lead to stability and stabilizability? The thesis initiates a systematic inquiry into this question and provides rigorous justifications.Firstly, coupled linear systems and coupled nonlinear systems are investigated. Necessary and sufficient conditions in terms of the connectivity of the interaction directed graphs are derived to ensure that the equilibrium subspace is (globally uniformly) at tractive for systems with both fixed and dynamic interaction structures. We apply the results to several analysis and control synthesis problems including problems in synchronization of coupled Kuramoto oscillators, biochemical reaction network, and synthesis of rendezvous controllers for multi-agent systems. Secondly, the stabilizability problem of coupled kinematic unicycles is investigated when only local information is available. Necessary and sufficient graphical conditions are obtained to determine the feasibility of certain formations (point formations and line formations). Furthermore, we show that under certain graphical condition, stabilization of the vehicles to any geometric formation is also feasible provided the vehicles have a common sense of direction.