  Networks are a widely used and efficient paradigm to model real-world systems
where basic units interact pairwise. Many body interactions are often at play,
and cannot be modelled by resorting to binary exchanges. In this work, we
consider a general class of dynamical systems anchored on hypergraphs.
Hyperedges of arbitrary size ideally encircle individual units so as to account
for multiple, simultaneous interactions. These latter are mediated by a
combinatorial Laplacian, that is here introduced and characterised. The
formalism of the Master Stability Function is adapted to the present setting.
Turing patterns and the synchronisation of non linear (regular and chaotic)
oscillators are studied, for a general class of systems evolving on
hypergraphs. The response to externally imposed perturbations bears the imprint
of the higher order nature of the interactions.
