  This paper proves local-global principles for Galois cohomology groups over
function fields $F$ of curves that are defined over a complete discretely
valued field. We show in particular that such principles hold for $H^n(F,
Z/mZ(n-1))$, for all $n>1$. This is motivated by work of Kato and others, where
such principles were shown in related cases for $n=3$. Using our results in
combination with cohomological invariants, we obtain local-global principles
for torsors and related algebraic structures over $F$. Our arguments rely on
ideas from patching as well as the Bloch-Kato conjecture.
