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It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree > 1, which is locally homogeneous of degree k with respect to a local Euler eld) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the rst result relies on the characterization of invariant di erential forms with respect to the graded Lie algebra of in nitesimal automorphisms, and on the study of the local properties of Hamiltonian vector elds on locally multisymplectic manifolds. In particular it is proved that the group of multisymplectic di eomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of in nitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic di eomorphisms.