Crustal shortening and thickening to c. 70–85 km in the Tibetan Plateau occurred both before and mainly after the c. 50 Ma India–Asia collision. Potassic–ultrapotassic shoshonitic and adakitic lavas erupted across the Qiangtang (c. 50–29 Ma) and Lhasa blocks (c. 30–10 Ma) indicate a hot mantle, thick crust and eclogitic root during that period. The progressive northward underthrusting of cold, Indian mantle lithosphere since collision shut off the source in the Lhasa block at c. 10 Ma. Late Miocene–Pleistocene shoshonitic volcanic rocks in northern Tibet require hot mantle. We review the major tectonic processes proposed for Tibet including ‘rigid-block', continuum and crustal flow as well as the geological history of the major strike-slip faults. We examine controversies concerning the cumulative geological offsets and the discrepancies between geological, Quaternary and geodetic slip rates. Low present-day slip rates measured from global positioning system and InSAR along the Karakoram and Altyn Tagh Faults in addition to slow long-term geological rates can only account for limited eastward extrusion of Tibet since Mid-Miocene time. We conclude that despite being prominent geomorphological features sometimes with wide mylonite zones, the faults cut earlier formed metamorphic and igneous rocks and show limited offsets. Concentrated strain at the surface is dissipated deeper into wide ductile shear zones.