  We develop semistrict higher gauge theory from first principles. In
particular, we describe the differential Deligne cohomology underlying
semistrict principal 2-bundles with connective structures. Principal 2-bundles
are obtained in terms of weak 2-functors from the Cech groupoid to weak Lie
2-groups. As is demonstrated, some of these Lie 2-groups can be differentiated
to semistrict Lie 2-algebras by a method due to Severa. We further derive the
full description of connective structures on semistrict principal 2-bundles
including the non-linear gauge transformations. As an application, we use a
twistor construction to derive superconformal constraint equations in six
dimensions for a non-Abelian N=(2,0) tensor multiplet taking values in a
semistrict Lie 2-algebra.
