  Following the same steps made for a scalar field in a parallel publication,
we propose a class of perturbative theories of quantum gravity based on
fractional operators, where the kinetic operator of the graviton is either made
of fractional derivatives or a covariant fractional d'Alembertian. The
classical action for each theory is constructed and the equations of motion are
derived. Unitarity and renormalizability of theories with a fractional
d'Alembertian are also considered. We argue that unitarity and power-counting
renormalizability never coexist, although in some cases one-loop unitary and
finiteness are possible. One of the theories is unitary and infrared-finite and
can serve as a ghost-free model with large-scale modifications of general
relativity.
