Let G be a reductive group scheme over a regular ring R. Recently,
there have been lots of progress regarding the structure of G-torsors.
To give one example, an old conjecture of Grothendieck and Serre
predicts that if R is local, then any G-torsor is trivial, provided it
is trivial over the fraction field of R. This is now a theorem when R
contains a field. The mixed characteristic story is rapidly
developing. The proofs are inspired by many areas of mathematics such
as A1 homotopy theory and affine Grassmannians.
After a brief introduction to torsors, I will discuss the current
state of Grothendieck-Serre and related conjectures, as well as the
main proof techniques.