  We study the notion of duality in the context of graded manifolds. For graded
bundles, somehow like in the case of Gelfand representation and the duality:
points vs. functions, we obtain natural dual objects which belongs to a
different category than the initial ones, namely graded polynomial (co)algebra
bundles and free graded Weil (co)algebra bundles. Our results are then applied
to obtain elegant characterizations of double vector bundles and graded bundles
of degree 2. All these results have their supergeometric counterparts. For
instance, we give a simple proof of a nice characterisation of $N$-manifolds of
degree 2, announced in the literature.
