  The descent method is one of the strategies allowing one to study the
Brauer--Manin obstruction to the local--global principle and to weak
approximation on varieties over number fields, by reducing the problem to
``descent varieties''. Very recently, in his Park City lecture notes,
Wittenberg formulated a ``descent conjecture'' for torsors under linear
algebraic groups. The present article gives a proof of this conjecture in the
case of connected groups, generalizing the toric case from the previous work of
Harpaz--Wittenberg. As an application, we deduce directly from Sansuc's work
the theorem of Borovoi for homogeneous spaces of connected linear algebraic
groups with connected stabilizers. We are also able to reduce the general case
to the case of finite (\'etale) torsors. When the set of rational points is
replaced by the Chow group of zero-cycles, an analogue of the above conjecture
for arbitrary linear algebraic groups is proved.

Algebraic Geometry;Number Theory