  We consider the possibility that some supersymmetric gauge theories which are
not asymptotically free can be governed by ultraviolet-stable fixed points. If
this scenario can be realized, gaugino masses will exhibit power-law running
with scale, providing a possible solution to the supersymmetric flavor problem.
While naive perturbative calculations hint at the appearance of
ultraviolet-stable fixed points in certain theories, there are strong
constraints following from limits on the scaling dimensions of gauge-invariant
operators, positivity of central charges, and Cardy's conjectured constraint on
the flow of the Euler coefficient in the stress tensor trace anomaly. Also, we
prove that if ultraviolet-stable fixed points do exist, they cannot occur in
the perturbative regime of a renormalizable model, and that all-orders results
in the limit of large numbers of chiral superfields are necessarily
inconclusive. However, we argue that the general idea of ultraviolet-stable
fixed points in supersymmetric gauge theories is viable, and exhibit models
that can satisfy all known constraints.
