  Nonlocal quantum corrections to gravity have been recently proposed as a
possible solution to the cosmological fine tuning problems. We study the
dynamics of a class of nonlocal actions defined by a function of the inverse
d'Alembertian of the Ricci scalar. Power-law and expnential functions are
considered in detail, but we also show a method to reconstruct a nonlocal
correction that generates a given background expansion. We find that even the
simplest terms can, while involving only Planck scale constants, drive the late
time acceleration without changing early cosmology. This leads to a sudden
future singularity, which however may be avoided by regularizing the
d'Alembertian. We also consider the Einstein frame versions of these models.
