  This document is an expanded version of a lecture presented at a conference
on "Thin Groups and Superstrong Approximation" held at the Mathematical
Sciences Research Institute in February 2012. Superstrong approximation is a
criterion on a finitely generated group, saying that certain Cayley graphs
associated to finite quotients of the group form an expander family. In recent
years, our knowledge about superstrong approximation for infinite-index
Zariski-dense subgroups of arithmetic lattices ("thin groups") has drastically
improved. We briefly survey the construction of monodromy groups, discuss our
(limited) knowledge about whether such groups are thin, and discuss an
application to arithmetic geometry (see the paper "Expander graphs, gonality,
and variation of Galois representations") deriving from recent advances in
superstrong approximation. We conclude by indulging in some speculations about
more general contexts, asking: what are the interesting questions about
"nonabelian superstrong approximation" and "superstrong approximation for
Galois groups?" We discuss the relation of these notions with the Product
Replacement Algorithm and the Bogomolov property for infinite algebraic
extensions of number fields.
