  The $N=2$ supersymmetric {\it self-dual} Yang-Mills theory and the $N=4$ and
$N=2$ {\it self-dual} supergravities in $2+2$ space-time dimensions are
formulated for the first time. These formulations utilize solutions of the
Bianchi identities subject to the super-Yang-Mills or supergravity constraints
in the relevant $N$-extended superspace with the space-time signature $(2,2)$.
