  The equivariant coarse index is well-understood and widely used for actions
by discrete groups. We extend the definition of this index to general locally
compact groups. We use a suitable notion of admissible modules over
$C^*$-algebras of continuous functions to obtain a meaningful index. Inspired
by work by Roe, we then develop a localised variant, with values in the
$K$-theory of a group $C^*$-algebra. This generalises the Baum-Connes assembly
map to non-cocompact actions. We show that an equivariant index for
Callias-type operators is a special case of this localised index, obtain
results on existence and non-existence of Riemannian metrics of positive scalar
curvature invariant under proper group actions, and show that a localised
version of the Baum-Connes conjecture is weaker than the original conjecture,
while still giving a conceptual description of the $K$-theory of a group
$C^*$-algebra.
