  We study gravitational lensing by compact objects in gravity theories that
can be written in a Post-Post-Newtonian (PPN) framework: i.e., the metric is
static and spherically symmetric, and can be written as a Taylor series in m/r,
where m is the gravitational radius of the compact object. Working invariantly,
we compute corrections to standard weak-deflection lensing observables at first
and second order in the ratio of the angular gravitational radius to the
angular Einstein ring radius of the lens. We show that the first-order
corrections to the total magnification and centroid position vanish universally
for gravity theories that can be written in the PPN framework. This arises from
some surprising, fundamental relations among the lensing observables in PPN
gravity models. We derive these relations for the image positions,
magnifications, and time delays. A deep consequence is that any violation of
the universal relations would signal the need for a gravity model outside the
PPN framework (provided that some basic assumptions hold). In practical terms,
the relations will guide observational programs to test general relativity,
modified gravity theories, and possibly the Cosmic Censorship conjecture. We
use the new relations to identify lensing observables that are accessible to
current or near-future technology, and to find combinations of observables that
are most useful for probing the spacetime metric. We give explicit applications
to the Galactic black hole, microlensing, and the binary pulsar J0737-3039.
