  A group may be considered $C^*$-stable if almost representations of the group
in a $C^*$-algebra are always close to actual representations. We initiate a
systematic study of which discrete groups are $C^*$-stable or only stable with
respect to some subclass of $C^*$-algebras, e.g. finite dimensional
$C^*$-algebras. We provide criteria and invariants for stability of groups and
this allows us to completely determine stability/non-stability of
crystallographic groups, finitely generated torsion-free step-2 nilpotent
groups, surface groups, virtually free groups and certain Baumslag-Solitar
groups.
