There is much speculation and a modest amount of evidence that certain mesons might form quasi-bound states with nuclei to produce really exotic states of matter.For this to be a practical possibility, the interaction between the meson and nucleons at low energies must be strong and attractive and the production rates "healthy".The conditions for this are surveyed for the K, η, ω, η , and φ mesons.How this might lead to quasi-bound states is then discussed in a few typical cases.Though some interesting effects have been seen in above-threshold data, the search for experimental signals for these exotic states with different mesons in bound state regions has generally been rather frustrating, with positive claims only being made for the η and the K -.