  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.
