  We address the stability issue of Ricci-flat and maximally symmetric
spacetimes in nonlocal gravity to all perturbative orders in the gravitational
perturbation. Assuming a potential at least cubic in curvature tensors but
quadratic in the Ricci tensor, our proof consists on a mapping of the stability
analysis in nonlocal gravity to the same problem in Einstein-Hilbert theory.
One of the consequences is that only the graviton field can propagate and the
theory is ghost-free at all perturbative orders. All the results known in
Einstein gravity in vacuum with or without a cosmological constant can be
exported to the case of nonlocal gravity: if a spacetime is stable at all
perturbative orders in Einstein gravity, it is stable also in nonlocal gravity.
Minkowski and de Sitter spacetimes are particular examples. We also study how
the theory affects the propagation of gravitational waves in a cosmological
background.
