  Surveying some of the recent developments on approximate subgroups and
super-strong approximation for thin groups, we describe the Bourgain-Gamburd
method for establishing spectral gaps for finite groups and the proof of the
classification of approximate subgroups of semisimple algebraic groups over
finite fields. We then give a proof of the super-strong approximation for mod
$p$ quotients via random matrix products and a quantitative version of strong
approximation. Some applications to the group sieve are also presented.
