The role of the internal model principle is investigated in this paper for the coordination of relative-degree-one and relative-degree-two nonlinear systems. For relative-degree-one systems that are incrementally (output-feedback) passive, we propose internal-model-based distributed control laws which guarantee output synchronization to an invariant manifold driven by autonomous synchronized internal models. For relative-degree-two systems, we consider a different internal-model-based distributed control framework for solving a formation control problem where the agents have to track a reference signal available only to the leader agent. In both cases, the local controller is also able to reject the disturbance signals generated by a local exosystem.