The notion of moment map is central in symplectic geometry, where thefunctions on the symplectic manifold (the "observables") form a Liealgebra. We extend this notion to higher differential forms, defining a homotopy moment map to be an $L_{\infty}$-algebra morphism into the observables. We give an interpretation in terms of coboundaries in a certain double complex, show that certain equivariant cocycles induce homotopy moment maps, and discuss obstructions and the relation to loop spaces. This is joint work in progress with Chris L. Rogers (Göttingen) and Yael Fregier (MIT).