This talk, which reports on joint work with Julia Hartmann and Daniel Krashen, concerns local-global principles over function fields of curves that are defined over a complete discretely valued field. The analogous situation for curves over a finite field is more classical. There, for any linear algebraic group G, the Tate- Shafarevich set (which is a group if G is commutative) measures the obstruction to a local-global principle for G-torsors; and this is always finite. This talk will present finiteness theorems for Tate-Shafarevich sets in our situation, and will give necessary and sufficient conditions for this obstruction to vanish under appropriate hypotheses. As a consequence, we obtain local-global results for quadratic forms and central simple algebras.