  A group $G$ is called $W^*$-superrigid (resp. $C^*$-superrigid) if it is
completely recognizable from its von Neumann algebra $L(G)$ (resp. reduced
$C^*$-algebra $C_r^*(G)$). Developing new technical aspects in Popa's
deformation/rigidity theory we introduce several new classes of
$W^*$-superrigid groups which appear as direct products, semidirect products
with non-amenable core and iterations of amalgamated free products and
HNN-extensions. As a byproduct we obtain new rigidity results in $C^*$-algebra
theory including additional examples of $C^*$-superrigid groups and explicit
computations of symmetries of reduced group $C^*$-algebras.
