  Given a double vector bundle $D\to M$, we define a bigraded `Weil algebra'
$\mathcal{W}(D)$, which `realizes' the algebra of smooth functions on the
supermanifold $D[1,1]$. We describe in detail the relations between the Weil
algebras of $D$ and those of the double vector bundles $D',\ D"$ obtained by
duality operations. In particular, we show that double-linear Poisson
structures on $D$ can be described alternatively as Gerstenhaber brackets on
$\mathcal{W}(D)$, vertical differentials on $\mathcal{W}(D')$, or horizontal
differentials on $\mathcal{W}(D")$. We also give a new proof of Voronov's
result characterizing double Lie algebroid structures. In the case that $D=TA$
is the tangent prolongation of a Lie algebroid, we find that $\mathcal{W}(D)$
is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad-Crainic.
We show that the deformation complex of Lie algebroids, the theory of IM forms
and IM multivector fields, and 2-term representations up to homotopy all have
natural interpretations in terms of our Weil algebras.
