Plane strain-elastic wave propagation is studied for two dissimilar half spaces joined together at a plane interface by an elastic bond. The bond thickness is assumed to be small compared to the wavelength, and an appropriately simplified description of the bond behavior is introduced. Attention is focused on solutions corresponding to the propagation of interface waves along the bond. The existence of interface waves is found to be governed by a parameter involving bond stiffness and wavelength. The limiting case of an infinitely stiff bond corresponds to the interface wave problem first solved by Stoneley, and it is shown that the present analysis yields Stoneley’s frequency equation in this limit. Also, the limiting case of an infinitely soft bond is found as expected to give two Rayleigh surface waves, one in each medium. It is shown analytically that, for intermediate bond stiffnesses, there may occur zero, one, or two interface waves, depending on the properties of the bond and the media. Illustrative numerical examples are presented. It is the conclusion of this study that account must be taken of the stiffness of the bond and the wavelength of the disturbance before it is proper to speak of an interface wave existing or not existing at a bonded interface.