  We examine local cohomology in the setting of valuation rings. The novelty of
this investigation stems from the fact that valuation rings are usually
non-Noetherian, whereas local cohomology has been extensively developed mostly
in a Noetherian setting. We prove various vanishing results on local cohomology
for valuation rings of finite Krull dimension. These vanishing results stem
from a uniform bound on the global dimension of such rings. Our investigation
reveals differences in the sheaf theoretic definition of local cohomology, and
the algebraic definition in terms of a limit of certain Ext functors.
