  The aim of this article is to give an overview of several types of
Paley-Wiener theorems occuring in harmonic analysis related to symmetric
spaces. This will serve as a motivation for the introduction of the
$\Theta$-spherical functions, the correspondint $\Theta$-spherical Fourier
transform, and the Paley-Wiener theorem for this transform. Up to now such a
theorem has only been proven in very special cases, and still, its formulation
and proof are very technical. In this paper we do not go into details of the
proofs, but present an overview which explains the different examples which
have inspired and motivated the theory of $\Theta$-spherical functions.

Representation Theory;Functional Analysis