  We calculate the $\beta$-functions for $SO(N)$ and $SU(N)$ gauge theories
coupled to adjoint and fundamental scalar representations, correcting
long-standing, previous results. We explore the constraints on $N$ resulting
from requiring asymptotic freedom for all couplings. When we take into account
the actual allowed behavior of the gauge coupling, the minimum value of $N$ in
both cases turns out to be larger than realized in earlier treatments. We also
show that in the large $N$ limit, both models have large regions of parameter
space corresponding to total asymptotic freedom.
