  The decays K_{L,S} -> \pi^o \pi^o \nu \bar{\nu} and K_{L,S} -> \pi^+ \pi^-
\nu \bar{\nu} involve the weak transition from a strange to a non-strange
quark. Although they have considerably smaller branching ratios than those for
the corresponding rare processes involving single pions, some may be more
distinguishable experimentally from background processes and thus could provide
another probe of the s -> d \nu \bar{\nu} transition. Using recent knowledge of
the CKM matrix elements and measurements of related processes, we give improved
predictions for their branching ratios both within and beyond the Standard
Model.
