Given a field F and a set T of overfields F_t (t in T) of F, the local-global principle holds for a family C of algebraic varieties defined over F, if for any variety X in C, the existence of rational points on X over each F_t implies the existence of a rational point on X over F.

In this talk, we discuss the validity of this property for families of principal homogenous spaces of linear algebraic groups G over a semiglobal field F. By semi-global field we mean the field F of rational functions of a curve Y defined over the field of fractions of a complete discrete valuation ring R. An original case of interest is that of curves over a p-adic field. One natural family F_t is that of completions at all discrete valuations of F.

The results depend on the reduction properties of the curve Y and of the group G, and on the arithmetic of the residue field of the valuation ring R.

We give a counterexample to the local-global principle for a semisimple simply connected group G. This is in sharp contrast with the classical number theory situation.

The talk will start with a quick survey and then describe recent joint work with D. Harbater, J. Hartmann, D. Krashen, R. Parimala and V. Suresh.