To handle the common existing constraints, that is, limited energy supplies and limited communication bandwidth in multiagent systems (MASs), this article investigates the consensus problem in MASs with event-triggered communication (ETC) and state quantization. In order to compensate for the effect brought by mismatched disturbances, we also propose a novel multiple discontinuous sliding-mode surface, and the corresponding sliding-mode control law is constructed by considering the event-triggered and dynamic quantized mechanisms jointly. Under such a scheme, it is shown that the state trajectories of all the agents will be regulated to achieve consensus asymptotically and the Zeno behavior can be avoided completely. We further extend this work to self-triggered and periodic event-triggered cases. Particularly, in a periodic event-triggered approach, the new form of triggering conditions and upper bound of the sampling periods are provided explicitly. As a result, all agents can reach bounded consensus. Moreover, the upper bound of the consensus error can be arbitrarily adjusted by appropriately selecting parameters, and the periodic event-triggered case will be reduced to the event-triggered case when the bound approaches 0 (sampling periods approach 0 at the same time). A numerical example is illustrated to verify the effectiveness of the proposed algorithms.