  In this paper we summarize and give examples of a generalization of the
coorbit space theory initiated in the 1980's by H.G. Feichtinger and K.H.
Gr\"ochenig. Coorbit theory has been a powerful tool in characterizing Banach
spaces of distributions with the use of integrable representations of locally
compact groups. Examples are a wavelet characterization of the Besov spaces and
a characterization of some Bergman spaces by the discrete series representation
of $\mathrm{SL}_2(\mathbb{R})$. We present examples of Banach spaces which
could not be covered by the previous theory, and we also provide atomic
decompositions for an example related to a non-integrable representation.
