Abstract: With the deep understanding towards the objective laws of nature, requirements on refinement and complexity in engineering system design are increasing. Many coupled dynamic system designs need to take into account the dynamics induced by the time delay existing in the coupling process. Such coupling time delay may come from the process of coupling with the sensing system, the actuation system and the control system. Coupling delays also extensively exist in the fields such as transportation system, system biology, electronic communication, neural and information networks and etc. Firstly, based on the concept of coupling delay, this paper reviews the recent research progresses on dynamics induced by such delay from the following four aspects: (1) the delay-centered mechanism of complex dynamics in coupled systems; (2) experimental foundation and realization of stabilizing coupled systems by utilizing time delay; (3) dynamics of fast-slow coupled system with time delay; and (4) synchronization and desynchronization of delayed neural networks. Some advances in the general theory of systems with coupling delay are highlighted including the coupling-delay-induced bifurcation and singularity with high codimention and the novel quantitative method of analysis, normal form computation for neutral delay differential equations, identification of time delay and nonlinear parameters in nonlinear systems with coupling delay and the relevant experiment, relaxation oscillation in the fast-slow system with coupling delay, and transition of modes of synchronization induced by coupling delay in network systems. Secondly, as for the application, some new results are presented in details such as the coupling-delay-induced chatter in grinding process and its mechanism, bifurcation with high codimension and complex dynamics induced by coupling delay in neural networks with inertial terms, and design and experiments of vibration absorber and isolator using coupling delay. Finally, some problems which are worthy of attention in near future are highlighted from perspectives of the general theory of systems with coupling delay and the potential applications.