  Wave-mechanical effects in gravitational lensing have long been predicted,
and with the discovery of populations of compact transients such as
gravitational wave events and fast radio bursts, may soon be observed. We
present an observer's review of the relevant theory underlying wave-mechanical
effects in gravitational lensing. Starting from the curved-spacetime scalar
wave equation, we derive the Fresnel-Kirchoff diffraction integral, and analyze
it in the eikonal and wave optics regimes. We answer the question of what makes
interference effects observable in some systems but not in others, and how
interference effects allow for complementary information to be extracted from
lensing systems as compared to traditional measurements. We end by discussing
how diffraction effects affect optical depth forecasts and lensing near
caustics, and how compact, low-frequency transients like gravitational waves
and fast radio bursts provide promising paths to open up the frontier of
interferometric gravitational lensing.
