The unfolding of a system represents in a single branching structure all its possible computations: it is the cornerstone both of semantical constructions and of efficient partial order verification techniques. In this paper we survey the contributions we elaborated in the last decade with Ugo Montanari and other colleagues, concerning the unfolding of graph transformation systems, and its use in the definition of a Winskel style functorial semantics and in the development of methodologies for the verification of finite and infinite state systems.