  In this contribution we report on theoretical studies of $\eta$ nuclear
quasi-bound states in few- and many-body systems performed recently by the
Jerusalem-Prague Collaboration [1-5]. Underlying energy-dependent $\eta N$
interactions are derived from coupled-channel models that incorporate the
$N^*(1535)$ resonance. The role of self-consistent treatment of the strong
energy dependence of subthreshold $\eta N$ amplitudes is discussed. Quite large
downward energy shift together with rapid decrease of the $\eta N$ amplitudes
below threshold result in relatively small binding energies and widths of the
calculated $\eta$ nuclear bound states. We argue that the subthreshold behavior
of $\eta N$ scattering amplitudes is crucial to conclude whether $\eta$ nuclear
states exist, in which nuclei the $\eta$ meson could be bound and if the
corresponding widths are small enough to allow detection of these $\eta$
nuclear states in experiment.
