  The coalescence of double neutron star (NS-NS) and black hole (BH)-NS
binaries are prime sources of gravitational waves (GW) for Advanced LIGO/Virgo
and future ground-based detectors. Neutron-rich matter released from such
events undergo rapid neutron capture (r-process) nucleosynthesis as it
decompresses into space, enriching our universe with rare heavy elements like
gold and platinum. Radioactive decay of these unstable nuclei powers a rapidly
evolving, approximately isotropic thermal transient known as a ``kilonova',
which probes the physical conditions during the merger and its aftermath. Here
I review the history and physics of kilonovae, leading to the current paradigm
of day-timescale emission at optical wavelengths from lanthanide-free
components of the ejecta, followed by week-long emission with a spectral peak
in the near-infrared (NIR). These theoretical predictions, as compiled in the
original version of this review, were largely confirmed by the transient
optical/NIR counterpart discovered to the first NS-NS merger, GW170817,
discovered by LIGO/Virgo. Using a simple light curve model to illustrate the
essential physical processes and their application to GW170817, I then
introduce important variations about the standard picture which may be
observable in future mergers. These include ~hours-long UV precursor emission,
powered by the decay of free neutrons in the outermost ejecta layers or
shock-heating of the ejecta by a delayed ultra-relativistic outflow; and
enhancement of the luminosity from a long-lived central engine, such as an
accreting BH or millisecond magnetar. Joint GW and kilonova observations of
GW170817 and future events provide a new avenue to constrain the astrophysical
origin of the r-process elements and the equation of state of dense nuclear
matter.
