The Central Andes are the Earth's highest mountain belt formed by ocean-continent collision. Most of this uplift is thought to have occurred in the past 20 Myr, owing mainly to thickening of the continental crust, dominated by tectonic shortening. Here we use P-to-S (compressional-to-shear) converted teleseismic waves observed on several temporary networks in the Central Andes to image the deep structure associated with these tectonic processes. We find that the Moho (the Mohorovicić discontinuity--generally thought to separate crust from mantle) ranges from a depth of 75 km under the Altiplano plateau to 50 km beneath the 4-km-high Puna plateau. This relatively thin crust below such a high-elevation region indicates that thinning of the lithospheric mantle may have contributed to the uplift of the Puna plateau. We have also imaged the subducted crust of the Nazca oceanic plate down to 120 km depth, where it becomes invisible to converted teleseismic waves, probably owing to completion of the gabbro-eclogite transformation; this is direct evidence for the presence of kinetically delayed metamorphic reactions in subducting plates. Most of the intermediate-depth seismicity in the subducting plate stops at 120 km depth as well, suggesting a relation with this transformation. We see an intracrustal low-velocity zone, 10-20 km thick, below the entire Altiplano and Puna plateaux, which we interpret as a zone of continuing metamorphism and partial melting that decouples upper-crustal imbrication from lower-crustal thickening.