Over the last decades the theory of function spaces has moved from a special topic for a few experts to a broadly recognized area of mathematical analysis. Among others, function spaces have been a driving force for many application areas (Fourier analysis, PDEs, etc.), but also for the development of new and refined methods for linear functional analysis.

The presentation will focus on the different settings chosen by different groups and "schools", with a short discussion of the essence of (and difference between) such scenarios, be it the Luxemburg-Zaanen or the Triebel school, including some recent developments, such as the topic of ball function spaces.