We discuss some applications of WQOs to several fields were hierarchies and
reducibilities are the principal classification tools, notably to Descriptive
Set Theory, Computability theory and Automata Theory. While the classical
hierarchies of sets usually degenerate to structures very close to ordinals,
the extension of them to functions requires more complicated WQOs, and the same
applies to reducibilities. We survey some results obtained so far and discuss
open problems and possible research directions.Comment: 37 page