  We introduce differentiable stacks and explain the relationship with Lie
groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable
stacks. In particular, we establish the relationship between $S^1$-gerbes and
groupoid $S^1$-central extensions. We define connections and curvings for
groupoid $S^1$-central extensions extending the corresponding notions of
Brylinski, Hitchin and Murray for
  $S^1$-gerbes over manifolds. We develop a Chern-Weil theory of characteristic
classes in this general setting by presenting a construction of Chern classes
and Dixmier-Douady classes in terms of analogues of connections and curvatures.
We also describe a prequantization result for both $S^1$-bundles and
$S^1$-gerbes extending the well-known result of Weil and Kostant. In
particular, we give an explicit construction of $S^1$-central extensions with
prescribed curvature-like data.
