We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is $(3,4)$-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of $(3,4)$-trivial algebras and provide a number of examples.

2010 Mathematics Subject Classification: Primary 53C15; Secondary 22E25, 53C29, 53C30, 53C55, 53D20, 70G45.