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