We introduce and study the notion of representation up to homotopy of a
Lie algebroid, paying special attention to examples. We use representations up to homotopy
to define the adjoint representation of a Lie algebroid and show that the resulting
cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid
is defined and shown to coincide with Kalkman’s BRST model for equivariant cohomology
in the case of group actions. The relation of this algebra with the integration of Poisson and
Dirac structures is explained in .