  We study non-abelian differentiable gerbes over stacks using the theory of
Lie groupoids. More precisely, we develop the theory of connections on Lie
groupoid $G$-extensions, which we call "connections on gerbes", and study the
induced connections on various associated bundles. We also prove analogues of
the Bianchi identities. In particular, we develop a cohomology theory which
measures the existence of connections and curvings for $G$-gerbes over stacks.
We also introduce $G$-central extensions of groupoids, generalizing the
standard groupoid $S^1$-central extensions.
  As an example, we apply our theory to study the differential geometry of
$G$-gerbes over a manifold.
