  This article describes how the ideas promoted by the fundamental papers
published by M. Frazier and B. Jawerth in the eighties have influenced
subsequent developments related to the theory of atomic decompositions and
Banach frames for function spaces such as the modulation spaces and
Besov-Triebel-Lizorkin spaces.
  Both of these classes of spaces arise as special cases of two different,
general constructions of function spaces: coorbit spaces and decomposition
spaces. Coorbit spaces are defined by imposing certain decay conditions on the
so-called voice transform of the function/distribution under consideration. As
a concrete example, one might think of the wavelet transform, leading to the
theory of Besov-Triebel-Lizorkin spaces.
  Decomposition spaces, on the other hand, are defined using certain
decompositions in the Fourier domain. For Besov-Triebel-Lizorkin spaces, one
uses a dyadic decomposition, while a uniform decomposition yields modulation
spaces.
  Only recently, the second author has established a fruitful connection
between modern variants of wavelet theory with respect to general dilation
groups (which can be treated in the context of coorbit theory) and a particular
family of decomposition spaces. In this way, optimal inclusion results and
invariance properties for a variety of smoothness spaces can be established. We
will present an outline of these connections and comment on the basic results
arising in this context.
