Modulation spaces have been introduced in the early 80s. In contrast to Besov spaces, which are built on dyadic decompositions of the frequency domain, they make use uniform covering of the frequency domain. It was therefore natural to find a family of intermediate spaces. Since ordinary interpolation theory (for Banach spaces) does not seem to provide reasonable spaces in this context it was natural to resort to the geometric properties of the decompositions in the frequency domain. Together with Peter Groebner (PhD Vienna, 92) the family of alpha-modulation spaces has been designed, with a a dilation paramter $\apha$ from the interval $ [0,1]$. The case $alpha = 0$ corresponds to the ordinary modulation spaces (no dilation), while the limiting case $alpha = 1$ is related to wavelet theory and produces Besov spaces.

In the last few years alpha-modulation spaces have received a lot of attention, and results similar to the classical results of Frazier and Jawerth an others have been obtained recently, e.g. by Fornasier (+Fei), Borup and Nielsen. Many basic results concerning these spaces (even for the case of modulation spaces) have been derived using the even more general concept of decomposition spaces, which had a much more flexible construction in the background. So let us shortly recall this: The differnce between a Besov-space Bspq(Rd) and the corresponding modulation space Mspq(Rd) was - according to the design of those spaces - the change of the decomposition on the Fourier domain, while the norm in which the signal parts belong to certain compact subdomains are measured was the same Lp-norm, and moreover, the weight function is of the same order $s$ (governing the amount of smoothness) and some fine-tuning parameter $q$. The so-called theory of decomposition spaces (developed by Feichtinger and Groebner) was based on the idea that one can allow for much more general covering, that one can think of either continuous covering of (banana-like) "balls" which change shape and orientation not too much over the frequency domain. In order to use those decomposition one has to be abe to select an a good way discrete (so-called) admissible coverings, which do not have too much mutual overlap between neighbors. This, together with sufficiently smooth partitions of unity subordinate to the given coverings allows to define distributions spaces characterized by the behaviour of their pieces (expressed again by some weighted, perhaps mixed norm space).

Results about duality, Fourier multipliers or interpolation according to the complex method of interpolation, and of course the basic theory (including independence from the ingredients from the constituting auxiliary constructions, such as the "admissble partitions of unity" or the "equivalence class of coverings") have been already discussed in the early papers. Only very recently tight frames, Frazier-Jawerth type expansions or Banach frames for spaces of this type have been developed (e.g by Borup and Nielsen).