  For a proper action by a locally compact group $G$ on a manifold $M$ with a
$G$-equivariant Spin-structure, we obtain obstructions to the existence of
complete $G$-invariant Riemannian metrics with uniformly positive scalar
curvature. We focus on the case where $M/G$ is noncompact. The obstructions
follow from a Callias-type index theorem, and relate to positive scalar
curvature near hypersurfaces in $M$. We also deduce some other applications of
this index theorem. If $G$ is a connected Lie group, then the obstructions to
positive scalar curvature vanish under a mild assumption on the action. In that
case, we generalise a construction by Lawson and Yau to obtain complete
$G$-invariant Riemannian metrics with uniformly positive scalar curvature,
under an equivariant bounded geometry assumption.
