All Abelian p-form theories with gauge invariant equations of motion with at most second derivatives are classified, and the explicit actions are construction in all D < 12 dimensions. It is proven that these equations of motion can depend only on the field strengths of the p-forms. A previously undiscovered 4-form Galileon cubic theory is identified in D > 7.

Galileons originated as humble scalar field theories with a generalized shift symmetry, but have since enjoyed many generalizations (to vector fields, multi-Galileons, etc.). In order to have the correct number of (ghost-free) degrees of freedom, a sufficient condition is that the equations of motion are at most second order in derivatives. Here, it is shown how to achieve this for fields of arbitrary spin. It is still possible to go 'Beyond' such Galileon theories, as higher order equations of motion are allowed providing they are suitably degenerate.