A hypocentral trend surface is a continuous function of latitude and longititude fitted by least squares to a set of hypocenters so that it predicts depth to the ''middle'' of a Benioff Zone. In this paper we take a relatively simple approach to hypocential trend surface analysis. The hypocentral trend surface is constructed from a spherical surface harmonic expansion whose coefficients are selected so as to minimize the standard vertical deviation between hypocenters and trend surface. The vertical deviation of a hypocenter from the hypocentral trend surface is called its residual. Consideration of a hypocentral trend surface cannot be divorced from consideration of the associated residuals and their spatial distribution. Any analysis of Benioff Zone geometry trades-off presumed thickness of the zone with its presumed shape. This trade-off can be investigated by examining suites of candidate hypocentral trend surfaces (together with their associated residual distributions) generated by varying the number of degrees of freedom available to the trend surface. Hypocentral trend surfaces are generated for three high quality sets of hypocenters obtained by local seismic Honshu (Japan), Cook Inlet (Alaska) and South Peru. Hypocenters beneath Honshu generate a bimodal distribution of residuals about their trend surface. The vertical separation of the upper and lower sheets of this Double Benioff Zone averages about 33 km. No Double Benioff Zone configuration is discovered below Cook Inlet or South Peru. Regional trend surfaces are established for intermediate-depth teleseismic data from South and Middle America. In South America (0¿--40 ¿S) teleseismic data suggest that along-strike transitions between ''flat'' and moderately steeply dipping sections of the Benioff Zone are achieved by flexure of a coherent slab rather than by fragmentation of the slab into tear-bound flaps or separate tongues of lithosphere. Hypocentral trend surface analysis of teleseismic data from South Peru suggests a geometry similar to that indicated by hypocentral trend surface analysis of the local data of Hasegawa and Sacks. An eastward shallowing Benioff Zone beneath southwest Mexico is closely associated with the geometry of the trans-Mexican volcanic arc. The Benioff Zone beneath Central America is more steeply dipping. The flat slab-steep slab transition near 96 ¿W may relate to the difference in age of oceanic lithosphere on either side of the Tehuantepec Ridge. While the strong flexure of the subducted Nazca plate beneath Equator is closely associated with the Carnegie Ridge, the Nazca ridge extropolates slightly north of the flexure in subducted lithosphere beneath South Peru. The flexure appears instead more closely associated with the pronounced inflection in the map view shape of the leading edge of the overriding South American plate.