  We use deformation-rigidity theory in von Neumann algebra framework to study
probability measure preserving actions by wreath product groups. In particular,
we single out large families of wreath products groups satisfying various type
of orbit equivalence (OE) rigidity. For instance, we show that whenever $H$,
$K$, $\G$, $\La$ are icc, property (T) groups such that $H\wr \G$ is measure
equivalent to $K\wr \La$ then automatically $\G$ is measure equivalent to $\La$
and $H^{\G}$ is measure equivalent to $K^{\La}$. Rigidity results for von
Neumann algebras arising from certain actions of such groups (i.e.
W$^*$-rigidity results) are also obtained.
