We consider the possibility that instabilities of the Abrikosov-Suhl r
esonance lead to new fixed-point behavior of the Kondo effect in a lat
tice environment. In one scenario, a pairing component to the resonant
scattering develops in the Kondo lattice, leading to an odd-frequency
superconductor. We discuss experiments that can discriminate between
this picture and d-wave pairing, and its relationship to the non-Fermi
-liquid fixed point of the overscreened Kondo model.