SUMMARY Finite-fault, broad-band teleseismic P waveform inversion was applied to the large Mw 8.0 Jalisco, Mexico, earthquake of 1995 October 9. The earthquake hypocentre was located at shallow depths just near the boundary zone between two lithospheric plates, Rivera and Cocos, and was felt with a maximum intensity of 9 MM on the Pacific coast. It was the first earthquake of magnitude greater than 6 to occur in this region since the two great Jalisco earthquakes (Ms 8.0 and 7.6) of 1932. For inversion, we used vertical records from 14 digital stations situated 30d to 90d from the epicentre. We used the fixed rake of the thrust-type focal mechanism given by the preliminary Harvard CMT. The low-angle dipping fault plane was taken as a 180×90 km2 rectangle divided into 162 subfaults of 10×10 km2. Each subfault had five intermediate point sources along-strike, and five intermediate point sources downdip. The size of the fault plane was changed from an initial size of 110×60 km2, estimated by RESCO as an aftershock area, to the final size of 180 × 90 km2, which included all significant displacements along the fault. The hypocentre was embedded 40 km from the left edge of the 180 km long fault, 40 km down-dip from the top of the fault which was situated practically along the trench axis. A boxcar source time function of width 2.0 s was used for each discrete rupture interval of each subfault. The rupture velocity was taken to be 2.8 km s−1 which is approximately 80 per cent of the shear wave speed in the layer containing the hypocentre. The following results on the rupture history were obtained. (1) The rupture duration was about 55 s, and slip occurred within an area of about 180 × 90 km2 along the Middle American trench in the depth interval from 9 to 33 km. (2) There were two main stages in the rupture history: during the first 18 s two main asperities at a distance of about 30 km from the earthquake hypocentre were ruptured with maximum slip up to 610 cm at a depth of 12-15 km; then during the next 37 s rupturing was observed to the north along the trench. The slip distribution obtained shows that the faulting, originating near the boundary between the Rivera and Cocos plates, went mainly along the Rivera-North American plate boundary and is consistent with the subduction of the Rivera plate beneath North America.