  We study GIT quotients $X_\theta=V\!/\!\!/\!_\theta G$ whose linearisation
map defines an isomorphism between the group of characters of $G$ and the
Picard group of $X_\theta$ modulo torsion. Our main result establishes that the
Cox ring of $X_\theta$ is isomorphic to the semi-invariant ring of the
$\theta$-stable locus in $V$. This applies to quiver flag varieties, Nakajima
quiver varieties, hypertoric varieties, and crepant resolutions of threefold
Gorenstein quotient singularities with fibre dimension at most one. As an
application, we present a simple, explicit calculation of the Cox ring of the
Hilbert scheme of $n$-points in the affine plane.
