  We construct the Hodge dual for supermanifolds by means of the Grassmannian
Fourier transform of superforms. In the case of supermanifolds it is known that
the superforms are not sufficient to construct a consistent integration theory
and that the integral forms are needed. They are distribution-like forms which
can be integrated on supermanifolds as a top form can be integrated on a
conventional manifold. In our construction of the Hodge dual of superforms they
arise naturally. The compatibility between Hodge duality and supersymmetry is
exploited and applied to several examples. We define the irreducible
representations of supersymmetry in terms of integral and superforms in a new
way which can be easily generalised to several models in different dimensions.
The construction of supersymmetric actions based on the Hodge duality is
presented and new supersymmetric actions with higher derivative terms are
found. These terms are required by the invertibility of the Hodge operator.
