  We introduce and study measures and densities (= geometric measures) on
differentiable stacks, using a rather straightforward generalization of
Haefliger's approach to leaf spaces and to transverse measures for foliations.
In general we prove Morita invariance, a Stokes formula which provides
reinterpretations in terms of (Ruelle-Sullivan type) algebroid currents, and a
Van Est isomorphism. In the proper case we reduce the theory to classical
(Radon) measures on the underlying space, we provide explicit (Weyl-type)
formulas that shed light on Weinstein's notion of volumes of differentiable
stacks, in particular, in the symplectic case, we prove the conjecture left
open in Weinstein's "The volume of a differentiable stack". We also revisit the
notion of Haar systems (and the existence of cut-off functions). Our original
motivation comes from the study of Poisson manifolds of compact types, which
provide two important examples of such measures: the affine and the
Duistermaat-Heckman measures.
