  This survey article has two components. The first part gives a gentle
introduction to Serre's notion of $G$-complete reducibility, where $G$ is a
connected reductive algebraic group defined over an algebraically closed field.
The second part concerns consequences of this theory when $G$ is simple of
exceptional type, specifically its role in elucidating the subgroup structure
of $G$. The latter subject has a history going back about sixty years. We give
an overview of what is known, up to the present day. We also take the
opportunity to offer several corrections to the literature.
