We survey research on the homotopy theory of the space map(X, Y) consisting
of all continuous functions between two topological spaces. We summarize
progress on various classification problems for the homotopy types represented
by the path-components of map(X, Y). We also discuss work on the homotopy
theory of the monoid of self-equivalences aut(X) and of the free loop space LX.
We consider these topics in both ordinary homotopy theory as well as after
localization. In the latter case, we discuss algebraic models for the
localization of function spaces and their applications.