We consider Galois cohomology groups over function fields $F$ of curves that are defined over a complete discretely valued field. Motivated by work of Kato and others for $n=3$, we show that local-global principles hold for $H^n(F, {\mathbb Z}/m{\mathbb Z} (n-1))$ for all $n>1$. In the case $n=1$, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for $H^1(F,G)$, a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups $G$. Concrete applications of the results include central simple algebras and Albert algebras.