The quantification of the complexity of networks is, today, a fundamental problem in the physics of complex systems.  A possible roadmap to solve the problem is via extending key concepts of information theory to networks.  In this talk we discuss recent works defining the Shannon entropy of a network ensemble and evaluating how it relates to the Gibbs and von Neumann entropies of network ensembles.  The quantities we introduce here  play a crucial role for the formulation of null models of networks through maximum-entropy arguments and  contribute to inference problems emerging in the field of complex networks.