  We introduce an axiomatic framework for the parallel transport of connections
on gerbes. It incorporates parallel transport along curves and along surfaces,
and is formulated in terms of gluing axioms and smoothness conditions. The
smoothness conditions are imposed with respect to a strict Lie 2-group, which
plays the role of a band, or structure 2-group. Upon choosing certain examples
of Lie 2-groups, our axiomatic framework reproduces in a systematical way
several known concepts of gerbes with connection: non-abelian differential
cocycles, Breen-Messing gerbes, abelian and non-abelian bundle gerbes. These
relationships convey a well-defined notion of surface holonomy from our
axiomatic framework to each of these concrete models. Till now, holonomy was
only known for abelian gerbes; our approach reproduces that known concept and
extends it to non-abelian gerbes. Several new features of surface holonomy are
exposed under its extension to non-abelian gerbes; for example, it carries an
action of the mapping class group of the surface.
