**ABSTRACT:**

Decomposition spaces were introduced by Feichtinger and GrÃ¶bner in 1983, shortly after the introduction of modulation spaces. Prototypes of decomposition spaces are Banach spaces of functions which can be described via specific types of frequency decompositions (on the Fourier side, that is) like Besov spaces and modulation spaces, e.g.

While Feichtinger and GrÃ¶berner's original paper showed that decomposition spaces coincide if (among other conditions) their respective underlying decompositions are comparable, the recent work of Felix Voigtlaender has provided an extensive list of necessary and/or sufficient conditions for coincidence and embeddings of decomposition spaces. The distinctive feature of his work is the almost universal applicability within the realm of well-known decomposition spaces, which goes far beyond the usual case-by-case studies.

The aim of the talk is to give a brief introduction to the concepts and to illustrate Voigtlaender's results through classical, more recent and brandnew examples.