  Short history of ideas concerning a possible existence of bound states of the
$\eta$-meson and a nucleus is considered. First experiments at BNL and LAMPF on
searching for these states are discussed. Another recent experiment using the
photon beam of the 1 GeV electron synchrotron of LPI is described. Possible
experiments on studying $\eta$-mesic nuclei using a proton beam (at Nuclotron
of Dubna) and a $\gamma$-beam (CEBAF, JLAB) are suggested.
