This paper investigates the average-consensus problem of multi-agent systems with direct and weighted topologies.Event-triggered control laws are adopted so as to reduce the frequency of individual control updating since the agents may be resource-limited in many real systems.The discrete time instants where the events are triggered are determined by a trigger function with respect to a certain measurement error.A centralized average-consensus protocol is proposed first for networks with fixed interaction topology,the stability and influencing factors of which are also analyzed.The design of trigger functions for networks with variable topology is also discussed.Then the results are extended to the decentralized counterpart,in which agents require only the information of their neighbors.Numerical examples are also provided that demonstrate the effectiveness of the theoretical results. This paper investigates the average-consensus problem of multi-agent systems with direct and weighted topologies. Event-triggered control laws are being so as to reduce the frequency of individual control updating since the agents may be resource-limited in many real systems. discrete time instants where the events are triggered are with determined responses by a trigger function with respect to a certain measurement error. A centralized average-consensus protocol is proposed first for networks with fixed interaction topology, the stability and influencing factors of which are also analyzed. design of trigger functions for networks with variable topology is also discussed.Then the results are extended to the decentralized counterpart, in which agents require only the information of their neighbors. Numerical examples are also provided that demonstrate the effectiveness of the theoretical results.