A semitoric system is a four-dimensional integrable system whose momentum map has a component generating an effective S^1-action and possesses mild singularities. These systems have been classified by Pelayo and Vũ Ngọc thanks to several invariants, but the construction of a system with given invariants involves symplectic gluing of some local normal forms. My goal will be to review these notions and to explain some attempts to answer the following question: can one construct a fully explicit (i.e. whose momentum map is given by global explicit formulas) semitoric system with a given partial list of invariants in a relatively simple way? This will lead us to some more general systems with S^1-symmetry called hypersemitoric systems. This is based on past and ongoing work with J. Palmer (University of Illinois Urbana-Champaign).
