This article investigates the problem of event-triggered distributed model predictive control (DMPC) for continuous-time nonlinear multiagent systems (MASs) subject to bounded disturbances, input and communication delays simultaneously. The compensation schemes for input and communication delays are proposed, respectively. A new optimal control problem (OCP) to achieve consensus among all agents is formulated under the proposed delay compensation schemes. To alleviate the communication burden and sensing cost, a novel input delay-related periodic event-triggering condition is proposed based on the state error to determine when to calculate the new control input. Sufficient condition guaranteeing the feasibility of the OCP is achieved with the proposed event-triggered control scheme. In stability analysis, a novel time-varying Lyapunov function is constructed, and the input-to-state practical stability of the MASs is derived by using the quasi-Lipschitz continuous property of the Lyapunov function. Numerical results related to vehicle platooning demonstrate the effectiveness of the proposed event-triggered DMPC.