Abstract. This note is a survey of the main results known about fixed subgroups of endomorphisms of finitely generated free groups. A historic point of view is taken, emphasizing the evolution of this line of research, from its beginning to the present time. The article concludes with a section containing the main open problems and conjectures, with some comments and discussions on them.

An element g of a free group F depends on a subgroup H < F , or is algebraic over H, if it satisfies a univariate equation over H. Equivalently, when H is finitely-generated, the rank of H, g is at most the rank of H. We study the elements that depend on subgroups of F , the subgroups they generate and the equations that they satisfy.