The state observer for dynamic links in complex dynamical networks (CDNs) is investigated by using the adaptive method whether the networks are undirected or directed. In this paper, a complete network model is proposed, which is composed of two coupled subsystems called nodes subsystem and links subsystem, respectively. Especially, for the links subsystem, associated with some assumptions, the state observer with parameter adaptive law is designed. Compared to the existing results about the state observer design of CDNs, the advantage of this method is that a estimation problem of dynamic links is solved in directed networks for the first time. Finally, the results obtained in this paper are demonstrated by performing a numerical example.