I consider the possibility that supersymmetric gauge theories 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. Naive
perturbative calculations hint at the appearance of such
ultraviolet-stable fixed points in certain theories, but there are strong
constraints following from unitarity and from a conjectured constraint of
Cardy. However, I argue that the general idea of ultraviolet-stable fixed
points in supersymmetric gauge theories is indeed viable, and exhibit
simple models that can satisfy all known constraints.