  We consider parahoric Bruhat-Tits group schemes over a smooth projective
curve and torsors under them. If the characteristic of the ground field is
either zero or positive but not too small and the generic fiber is absolutely
simple and simply-connected, we show that such group schemes can be written as
invariants of reductive group schemes over a tame cover of the curve. We relate
the torsors under the Bruhat-Tits group scheme and torsors under the reductive
group scheme over the cover which are equivariant for the action of the
covering group. For this, we develop a theory of local types for such
equivariant torsors. We also relate the moduli stacks of torsors under the
Bruhat-Tits group scheme and equivariant torsors under the reductive group
scheme over the cover. In an Appendix, B. Conrad provides a proof of the Hasse
principle for adjoint groups over function fields with finite field of
constants.
