It is a long way since wavelet theory and time-frequency analysis have found a great deal of attention, as two different ways of decomposing signals in orthogonal or also non-orthogonal ways. The unifying theory, covering both cases, distilling from these two situations the common group theoretical background lead to the theory of coorbit spaces. Starting from an integrable and irreducible representation of some locally compact group (such as the "ax+b"-group or the Heisenberg group) one can derive families of Banach spaces having natural atomic characterizations, or alternatively a continuous transform associated to it.

While unification of these two groups, also making the analogy to Banach spaces of analytic functions invariant under the Moebius group have been at the heart of the orginal theory recent years have seen further new instances and generalizations. Among them shearlets or the Blaschke product should be mentioned here.

The talk will try to summarize a few of the general principles which can be derived from the general theory, but also highlight the difference between the different groups and signal expansions arising from corresponding group representations. There is still a lot to be done, also from the point of view of applications and the numerical realization of such non-orthogonal expansions.