In this paper, the observer-based consensus problem for nonlinear multi-agent systems is considered. The dynamics of each agent is given in general form of Lipschitz nonlinear system, and the communication topology among the agents is assumed to be undirected and connected. The leader-following case and leaderless case are discussed. In the former, it is assumed that the leader’s input is possibly nonzero and time-varying and only a subset of the following agents can access the state information of the leader. To track the active leader, a distributed adaptive consensus protocol, based on the relative-output information with its neighbouring agents, is proposed for each following agent. It is shown that under suitable conditions, all the following agents can track the leader under the designed adaptive controllers and observers. Following that, the leaderless case is probed. Finally, a numerical example is given to illustrate our obtained result.