  We introduce a notion of moment map adapted to actions of Lie groups that
preserve a closed three-form. We show existence of our multi-moment maps in
many circumstances, including mild topological assumptions on the underlying
manifold. Such maps are also shown to exist for all groups whose second and
third Lie algebra Betti numbers are zero. We show that these form a special
class of solvable Lie groups and provide a structural characterisation. We
provide many examples of multi-moment maps for different geometries and use
them to describe manifolds with holonomy contained in G_2 preserved by a
two-torus symmetry in terms of tri-symplectic geometry of four-manifolds.
