  In this article, we establish the Grothendieck-Serre conjecture over
valuation rings: for a reductive group scheme $G$ over a valuation ring $V$
with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over
$K$. This result is predicted by the original Grothendieck-Serre conjecture and
the resolution of singularities. The novelty of our proof lies in overcoming
subtleties brought by general nondiscrete valuation rings. By using flasque
resolutions and inducting with local cohomology, we prove a non-Noetherian
counterpart of Colliot-Th\'el\`ene-Sansuc's case of tori. Then, taking
advantage of techniques in algebraization, we obtain the passage to the
Henselian rank one case. Finally, we induct on Levi subgroups and use the
integrality of rational points of anisotropic groups to reduce to the
semisimple anisotropic case, in which we appeal to properties of parahoric
subgroups in Bruhat-Tits theory to conclude. In the last section, by using
properties of reflexive sheaves, we also prove a variant of Nisnevich's purity
conjecture.
