We define a construction operation on hypergraphs based on a colimit and show that its expressiveness is equal to the graph expressions of Bauderon and Courcelle. We also demonstrate that by closing a set of rewrite rules under graph construction we obtain a notion of rewriting equivalent to the double-pushout approach of Ehrig. The usefulness of our approach for the compositional modelling of concurrent systems is then demonstrated by giving a semantics of process graphs (corresponding to a process calculus with mobility) and of Petri nets. We introduce on the basis if a hypergraph construction, a method for the static analysis of process graphs, related to type systems.