The subgroup structure of simple algebraic groups has been investigated by Liebeck, Seitz, Stewart, Testerman and others. Liebeck and Seitz classified the maximal subgroups; in particular, they showed that a simple algebraic group has only finitely many conjugacy classes of maximal subgroups. In this talk I will describe a short alternative proof of this result, using some ideas from model theory and the theory
of G-complete reducibility.