Aside from the natural simplicity of the idea of scale space (``constant shape'')
one of the reasons why {\it wavelet theory} had an immediate impact was the fact
that already in the very first papers the ability of wavelets to
characterize the elements of many function spaces considered important at that time (namely
$\Lpsp$-spaces, Besov and Triebel-Lizorkin spaces) has been established by Y.~Meyer.
One can use the continuous wavelet transform or alternatively the wavelet
coefficients with respect to a ``good'' orthogonal wavelet basis.
Also the real Hardy space and its dual, the
BMO-space can be characterized via wavelet theory, thus establishing
the connection to Calderon-Zygmund operators. These are exactly the
operators which have a ``diagonally concentrated'' ' matrix representation
with respect to such wavelet bases. Various boundedness results
for such operators appear as quite natural under this perspective.d

With {\it modulation spaces}, introduced by the author already in the early 80's the
story went the other way around. First their characterization via Gabor expansions
was established, resp.\ via the short-time Fourier transform,
typically using weighted mixed norm conditions.
Only long after the basic properties of those spaces had been established
it became clear that they are well suited for a description of questions arising
in time-frequency analysis, in the theory of slowly time-variant channels (relevant
for mobile communication) or for the description of pseudo-differential operators
using the Kohn-Nirenberg or Weyl calculus or Sj\"strand's class. Compared to wavelet
analysis the time-frequency point-of-view allows to tackle similar
problems over general LCA (locally compact Abelian) groups, which is
not only interesting for the sake of generality, but also because
it provides a good setting for the discretization of pseudo-differential
operators, providing some insight into the possibility of using
finite-dimensional computational methods in order to approximate
problems arising in a continuous setting.

In the most simple setting one can use the {\it Banach Gelfand triple }
$\SOGTrRd$, consisting of the Segal algebra $\SORdN = \MiRdN$ (the
functions with ambiguity function in $\LiRtd$), the Hilbert space
$\LtRd$ and the dual space $\SOPRd$ of all tempered distributions
with bounded short-time Fourier transform.

{\it Coorbit theory} provides a variety of other groups where integrable
representations provide a corresponding family of Banach spaces
and the appropriate atomic decompositions. Shearlets and shearlet
spaces are a recent member of this family of coorbit spaces.

It is the author's belief that many more so-called {\it flexible atomic decompositions }
(typically Banach frames for families of Banach spaces) will play an important role
for the treatment of possible new classes of pseudo-differential operators and that
a better understanding of their properties and of the corresponding
atomic (or molecular) decompositions % associated with such classes
will contribute to progress in this field.