  Photometric redshifts are essential in studies of both galaxy evolution and
cosmology, as they enable analyses of objects too numerous or faint for
spectroscopy. The Rubin Observatory, Euclid, and Roman Space Telescope will
soon provide a new generation of imaging surveys with unprecedented area
coverage, wavelength range, and depth. To take full advantage of these
datasets, further progress in photometric redshift methods is needed. In this
review, we focus on the greatest common challenges and prospects for
improvement in applications of photo-$z$'s to the next generation of surveys:
  - Gains in $performance$ -- i.e., the precision of redshift estimates for
individual galaxies -- could greatly enhance studies of galaxy evolution and
some probes of cosmology.
  - Improvements in $characterization$ -- i.e., the accurate recovery of
redshift $distributions$ of galaxies in the presence of uncertainty on
individual redshifts -- are urgently needed for cosmological measurements with
next-generation surveys.
  - To achieve both of these goals, improvements in the scope and treatment of
the samples of spectroscopic redshifts which make high-fidelity photo-$z$'s
possible will also be needed.
  For the full potential of the next generation of surveys to be reached, the
characterization of redshift distributions will need to improve by roughly an
order of magnitude compared to the current state of the art, requiring progress
on a wide variety of fronts. We conclude by presenting a speculative evaluation
of how photometric redshift methods and the collection of the necessary
spectroscopic samples may improve by the time near-future surveys are
completed.
