In a laterally varying multi-layered elastic medium the amplitudes of reflected waves often cannot be correctly modelled by geometrical acoustics because of the presence of caustics, shadows and diffractions. Maslov theory and the Gaussian beam method give correct amplitudes at caustics but do not give correct arrival times for waves diffracted from a corner on a reflector. The multi-fold path integral (MFPI) method presented in this paper is an alternative that addresses these problems. The MFPI method uses a combination of Kirchhoff-Helmholtz theory and geometrical acoustics to propagate waves between interfaces, and plane-wave reflection/transmission coefficients to propagate them across the interfaces. The generalized ray associated with an MFPI consists of a number of smooth segments (one segment across each layer) joined together at interfaces. The integrations are over the point at which this generalized ray touches each interface. We derive MFPI's for PP. SS and PS reflections and for a PP refraction. The resulting formulas are given in detail and from them the MFPIs for other seismic phases can easily be obtained by inspection. We also discuss how a given MFPI can be numerically evaluated in the frequency domain or in the time domain. For models in which velocities vary between interfaces, much time can be saved by the use of layer matrices that give travel times and spreading factors as functions of points on adjacent interfaces. Finally, the MFPI method is compared to other generalized ray theories.