  We construct an equivariant version of Ray-Singer analytic torsion for
proper, isometric actions by locally compact groups on Riemannian manifolds,
with compact quotients. We obtain results on convergence, metric independence,
vanishing for even-dimensional manifolds, a product formula, and a
decomposition of classical Ray-Singer analytic torsion as a sum over conjugacy
classes of equivariant torsion on universal covers. We do explicit computations
for the circle and the line acting on themselves, and for regular elliptic
elements of $\operatorname{SO}_0(3,1)$ acting on $3$-dimensional hyperbolic
space. Our constructions and results generalise several earlier constructions
of equivariant analytic torsion and their properties, most of which apply to
finite or compact groups, or to fundamental groups of compact manifolds acting
on their universal covers. These earlier versions of equivariant analytic
torsion were associated to finite or compact conjugacy classes; we allow
noncompact conjugacy classes, under suitable growth conditions.
