In this article, we address the event-triggered consensus problem for single-integrator multiagent systems by introducing a novel nonlinear control protocol. With the proposed consensus protocol, convergence with prescribed transient performance of the closed-loop system can be guaranteed while control execution is triggered by a delicately designed edge-based event-triggering mechanism. The protocol also ensures that no agent exhibits Zeno behavior. Moreover, the method can be extended to the design of a self-triggered algorithm that is able to determine the triggering time sequence without continuously accessing the network states.