I will present a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many non elementary problems, which occur naturally in logic, combinatorics, formal language, verification, etc., with complexities ranging from simple towers of exponentials to Ackermannian and beyond. I plan to skim over the following topics: subrecursive hierarchies: these are functions and classes that allow to define rapidly-growing functions by recursion over ordinal indices,¡/li¿ length function theorems for well-quasi-orders, which are the key combinatorial statements for the upper bounds of most of the non-elementary problems I will mention,¡/li¿ the definition of fast-growing complexity classes and some easy properties. The recent results will be taken from work together with S. Abriola, D. Figueira, S. Figueira, C. Haase, S. Haddad, and Ph. Schnoebelen.