  Electromagnetic and gravitational radiation do not propagate solely on the
null cone in a generic curved spacetime. They develop "tails," traveling at all
speeds equal to and less than unity. If sizeable, this off-the-null-cone effect
could mean objects at cosmological distances, such as supernovae, appear dimmer
than they really are. Their light curves may be distorted relative to their
flat spacetime counterparts. These in turn could affect how we infer the
properties and evolution of the universe or the objects it contains. Within the
gravitational context, the tail effect induces a self-force that causes a
compact object orbiting a massive black hole to deviate from an otherwise
geodesic path. This needs to be taken into account when modeling the
gravitational waves expected from such sources. Motivated by these
considerations, we develop perturbation theory for solving the massless scalar,
photon and graviton retarded Green's functions in perturbed spacetimes,
assuming these Green's functions are known in the background spacetime. In
particular, we elaborate on the theory in perturbed Minkowski spacetime in
significant detail; and apply our techniques to compute the retarded Green's
functions in the weak field limit of the Kerr spacetime to first order in the
black hole's mass and angular momentum. Our methods build on and generalizes
work appearing in the literature on this topic to date, and lays the foundation
for a thorough, first principles based, investigation of how light propagates
over cosmological distances, within a spatially flat inhomogeneous
Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) universe. This perturbative
scheme applied to the graviton Green's function, when pushed to higher orders,
may provide approximate analytic (or semi-analytic) results for the self-force
problem in the weak field limits of the Schwarzschild and Kerr black hole
geometries.
