We describe the classification of the maximal closed positive-dimensional
subgroups of the classical groups defined over
an algebraically closed field and indicate the role which representation
theory plays. We then consider similar classification questions for
exceptional type groups or for the associated finite groups of Lie type,
where
representation theory also plays a significant role. The oldest results
which we discuss were established by Dynkin in the 50's and the most
recent result
appeared in 2004. Finally, we mention some current work on related
questions in the representation
theory of semisimple algebraic groups.


[Invited by Dr. Laura Ciobanu]