  We consider supersymmetric non-Abelian gauge theories coupled to hyper
multiplets on five and six dimensional orbifolds, S^1/Z_2 and T^2/Z_N,
respectively. We compute the bulk and local fixed point renormalizations of the
gauge couplings. To this end we extend supergraph techniques to these orbifolds
by defining orbifold compatible delta functions. We develop their properties in
detail. To cancel the bulk one-loop divergences the bulk gauge kinetic terms
and dimension six higher derivative operators are required. The gauge couplings
renormalize at the Z_N fixed points due to vector multiplet self interactions;
the hyper multiplet renormalizes only non-Z_2 fixed points. In 6D the
Wess-Zumino-Witten term and a higher derivative analogue have to renormalize in
the bulk as well to preserve 6D gauge invariance.
