Using the lottery preparation, we prove that any strongly unfoldable cardinal
$\kappa$ can be made indestructible by all < $\kappa$ -closed $\kappa^+$ -preserving forcing. This degree of indestructibility, we prove, is the
best possible from this hypothesis within the class of < $\kappa$ -closed forcing. From a stronger hypothesis, however, we prove that the
strong unfoldability of $\kappa$ can be made indestructible by all < $\kappa$ -closed forcing. Such indestructibility, we prove, does not follow from
indestructibility merely by < $\kappa$ -directed closed forcing. Finally, we obtain global and universal forms
of indestructibility for strong unfoldability, finding the exact consistency
strength of universal indestructibility for strong unfoldability.