In the mid-1980s, theoreticians predicted possible exotic states of an eta (η) meson and a nucleus bound by the strong interaction, so-called mesic nucleus. This led to extensive experimental efforts aimed at discovering such unstable states as well as understanding the underlying interactions via η meson production reactions. As experiments yielded new data, more and more sophisticated theoretical models were developed to understand the fundamental η-nucleon and η-nucleus interactions. After more than 35 years of research in this field, several signals have been identified as possible indications of hypothetical η-mesic nuclei, but there is still no general agreement or clear confirmation of their existence. In the paper, we review the study of the η-mesic nuclei with an emphasis on recent research activities.