This talk is aimed at presenting an on-going research on Lie systems possessing Vessiot—Guldberg Lie algebras of Hamiltonian vector fields relative to different types of geometric structures, e.g., Poisson, k-symplectic, and multisymplectic structures. As a main novelty, I will focus upon the case of multisymplectic structures, namely non-degenerate and closed k-forms. I will present some techniques for constructing such Lie systems, and I will study a reduction process for them. This suggests us to use a quite general type of multisymplectic momentum map and a multisymplectic reduction procedure for multisymplectic volume forms. Results will be illustrated with examples of physical and mathematical interest.