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It is known that the Brauer--Manin obstruction to the Hasse principle isvacuous for smooth Fano hypersurfaces of dimension at least $3$ over any numberfield. Moreover, for such varieties it follows from a general conjecture ofColliot-Th\'el\`ene that the Brauer--Manin obstruction to the Hasse principleshould be the only one, so that the Hasse principle is expected to hold.Working over the field of rational numbers and ordering Fano hypersurfaces offixed degree and dimension by height, we prove that almost every suchhypersurface satisfies the Hasse principle provided that the dimension is atleast $3$. This proves a conjecture of Poonen and Voloch in every case exceptfor cubic surfaces.

The Hasse principle for random Fano hypersurfaces