  These notes were prepared for the MSRI hot topics workshop on superstrong
approximation (2012). We give a brief overview of the developments in the
theory, especially the fundamental expansion theorem. Applications to
diophantine problems on orbits of integer matrix groups, the affine sieve,
group theory, gonality of curves and Heegaard genus of hyperbolic three
manifolds, are given. We also discuss the ubiquity of thin matrix groups in
various contexts, and in particular that of monodromy groups.
