  We investigate the structure of the scalar mesons $f_0(975)$ and $a_0(980)$
within realistic meson-exchange models of the $\pi\pi$ and $\pi\eta$
interactions. Starting from a modified version of the J\"ulich model for
$\pi\pi$ scattering we perform an analysis of the pole structure of the
resulting scattering amplitude and find, in contrast to existing models, a
somewhat large mass for the $f_0(975)$ ($m_{f_0}=1015$ MeV, $\Gamma_{f_0}=30$
MeV). It is shown that our model provides a description of
$J/\psi\rightarrow\phi\pi\pi/\phi KK$ data comparable in quality with those of
alternative models. Furthermore, the formalism developed for the $\pi\pi$
system is consistently extended to the $\pi\eta$ interaction leading to a
description of the $a_0(980)$ as a dynamically generated threshold effect
(which is therefore neither a conventional $q\overline{q}$ state nor a
$K\overline{K}$ bound state). Exploring the corresponding pole position the
$a_0(980)$ is found to be rather broad ($m_{a_0}=991$ MeV, $\Gamma_{a_0}=202$
MeV). The experimentally observed smaller width results from the influence of
the nearby $K\overline{K}$ threshold on this pole.
