  The theory of principal $G$-bundles over a Lie groupoid is an important one,
unifying the various types of principal $G$-bundles, including those over
manifolds, those over orbifolds, as well as equivariant principal $G$-bundles.
In this paper, we study the differential geometry of these objects, including
connections and holonomy maps. We also introduce a Chern-Weil map for these
principal bundles and prove that the characteristic classes obtained coincide
with the universal characteristic classes.
  As an application, we recover the equivariant Chern-Weil map of Bott-Tu. We
also obtain an explicit chain map between the Weil model and the simplicial
model of equivariant cohomology which reduces to the Bott-Shulman map
$S({\mathfrak g}^*)^G \to H^*(BG)$ when the manifold is a point.
