Given a semigroup presentation P = ⟨ X | R ⟩ and a word u in X* the set of all reduced spherical (u,u)-semigroup diagrams over P (the analogue of Van Kampen diagrams for semigroup presentations) admits a `natural' structure of a group, called the diagram group over P based at u. Diagram groups were introduced by Meakin and Sapir, and the delightful theory of diagram groups ripened through the pioneering work of Kilibarda, Guba and Sapir. The talk is meant to be an introduction to the theory of diagram groups, with emphasis on diagram groups over trace monoids, a topic that I have been exploring together with Francesco Matucci.