Among seismic waves generated by widely dilferent types of sources, the waves that are the most interest- ing and of the longest duration are usually surface waves (which are also called normal waves, interference and channel waves, natural vibrations of layers, etc. ). These waves are distinguished by their dispersion and reso- nance, and are used in many investigations - in the determination of the structure of the medium (the existence of surface layers and waveguides), in the determination of the coordinates and properties of the source (in par- ticular its energy and mechanism), in the identification of subterranean exploSions, in the mapping of micro-- seisms, in the tracing of storms, etc. Outlines of the wide literature concerning surface waves can be found in 2, 9, 40J. Surface waves can be classed as Rayleigh or Love waves, depending on their velocity and polarization: the displacements of particles in Rayleigh waves are parallel, and the displacements in Love waves perpendicular, to the vertical plane containing the direction of propagation (some variation from this polarization is possible close to the source or in the presence of horizontal inhomogeneities in the medium). The present work is devoted to the calculation and investigation of Love waves in a vertically layered medium. The method used is based on the spectral theory of linear differential operators. ' This method is much more effective than previously applied methods based on the approximation of the medium by a set of homogeneous layers.