  We study local-global principles for torsors under reductive linear algebraic
groups over semi-global fields; i.e., over one variable function fields over
complete discretely valued fields. We provide conditions on the group and the
semiglobal field under which the local-global principle holds, and we compute
the obstruction to the local-global principle in certain classes of examples.
Using our description of the obstruction, we give the first example of a
semisimple simply connected group over a semi-global field where the
local-global principle fails. Our methods include patching and R-equivalence.
