  Metric theories of gravity are studied, beginning with a general action that
is quadratic in curvature and allows infinite inverse powers of the
d'Alembertian operator, resulting in infrared non-local extensions of general
relativity. The field equations are derived in full generality and their
consistency is checked by verifying the Bianchi identities. The weak-field
limit is computed and a straightforward algorithm is presented to infer the
post-Newtonian corrections directly from the action. We then apply this to
various infrared gravity models including non-local $Rf(R/ \Box)$ cosmology and
non-local dark energy and massive gravity models. Generically the Newtonian
potentials are not identical and deviate from the $1/r$ behaviour at large
distances. However, the former does not occur in a specific class of theories
that does not introduce additional degrees of freedom in flat spacetime. A new
nonlocal model within this class is proposed, defined by the exponential of the
inverse d'Alembertian. This model exhibits novel features, such as weakening of
the gravity in the infrared, suggesting de-gravitation of the cosmological
constant.
