The Mossbauer effect for atoms located at crystal surfaces is discussed. We show that the Einstein model is valid for describing qualitatively the mean square displacement of lattice atoms. The mean square displacement (Mossbauer recoilless fraction) and mean square velocity (Mossbauer second-order Doppler shift) of atoms in one and three-dimensional crystals with free surfaces are treated in a many-body approach. The simple models indicate that at high temperatures surface atom mean square displacements are on the order of twice bulk atom mean square displacements (or equivalently that surface atom Debye temperatures are about 1/square root of 2 times the corresponding bulk values) and that surface effects penetrate only a few atomic spacings into a crystal. A possible temperature dependence of the Mossbauer isomer shift due to anharmonic binding forces at surfaces is pointed out. The experimental difficulties faced in a surface Mossbauer study performed under ultrahigh vacuum conditions and the apparatus used in an attempt to overcome them are described. Mossbauer spectra of Fe(57) on LiF, in Ag, and on Ag have been obtained in the temperature range 100 to 400 K. (Author).