Abstract: The synchronization of multiplex complex dynamical networks with delayed coupling is investigated. The impact of topology on the synchronization of multiplex networks was studied. The design conditions required to achieve synchronization were determined by Lyapunov stability theory and LaSalle's invariance principle. We provided the control gain required for synchronization, achieving dynamic consistency in the motion trajectories of all node state variables within the multi-layer complex network, ultimately achieving dynamic synchronization of the entire multi-layer complex network. In the simulation, the effectiveness of the synchronization strategy for a three-layer complex network model with coupling terms and time delays was verified.