  $\beta$-functions for abelian and non-abelian gauge theories are studied in
the regime where the large $N$ flavor expansion is applicable. The first
nontrivial order in the 1/$N$ expansion is known for any value of $N\alpha$,
and there are also various indications as to the nature of higher order
effects. The singularity structure as a function of $N\alpha$ has implications
for the existence of nontrivial fixed points.
