  Connections and curvings on gerbes are beginning to play a vital role in
differential geometry and mathematical physics -- first abelian gerbes, and
more recently nonabelian gerbes. These concepts can be elegantly understood
using the concept of `2-bundle' recently introduced by Bartels. A 2-bundle is a
generalization of a bundle in which the fibers are categories rather than sets.
Here we introduce the concept of a `2-connection' on a principal 2-bundle. We
describe principal 2-bundles with connection in terms of local data, and show
that under certain conditions this reduces to the cocycle data for nonabelian
gerbes with connection and curving subject to a certain constraint -- namely,
the vanishing of the `fake curvature', as defined by Breen and Messing. This
constraint also turns out to guarantee the existence of `2-holonomies': that
is, parallel transport over both curves and surfaces, fitting together to
define a 2-functor from the `path 2-groupoid' of the base space to the
structure 2-group. We give a general theory of 2-holonomies and show how they
are related to ordinary parallel transport on the path space of the base
manifold.
