  Strong gravitational lensing of quasars has the potential to unlock the
poorly understood physics of these fascinating objects, as well as serve as a
probe of the lensing mass distribution and of cosmological parameters. In
particular, gravitational microlensing by compact bodies in the lensing galaxy
can enable mapping of quasar structure to $\lt 10^{-6}$ arcsec scales. Some of
this potential has been realized over the past few decades, however the
upcoming era of large sky surveys promises to bring this to full fruition. Here
we review the theoretical framework of this field, describe the prominent
current methods for parameter inference from quasar microlensing data across
different observing modalities, and discuss the constraints so far derived on
the geometry and physics of quasar inner structure. We also review the
application of strong lensing and microlensing to constraining the granularity
of the lens potential, i.e. the contribution of the baryonic and dark matter
components, and the local mass distribution in the lens, i.e. the stellar mass
function. Finally, we discuss the future of the field, including the new
possibilities that will be opened by the next generation of large surveys and
by new analysis methods now being developed.
