  We hereby study the properties of a large class of weakly nonlocal
gravitational theories around the (anti-) de Sitter spacetime background. In
particular, we explicitly prove that the kinetic operator for the graviton
field has the same structure as the one in Einstein-Hilbert theory around any
maximally symmetric spacetime. Therefore, the perturbative spectrum is the same
of standard general relativity, while the propagator on any maximally symmetric
spacetime is a mere generalization of the one from Einstein's gravity derived
and extensively studied in several previous papers. At quantum level the range
of theories here presented is superrenormalizable or finite when proper (non
affecting the propagator) terms cubic or higher in curvatures are added.
Finally, it is proven that for a large class of nonlocal theories, which in
their actions do involve neither the Weyl nor the Riemann tensor, the theory is
classically equivalent to the Einstein-Hilbert one with cosmological constant
by means of a metric field redefinition at any perturbative order.
