  Certain lattice wave systems in translationally invariant settings have one
or more spectral bands that are strictly flat or independent of momentum in the
tight binding approximation, arising from either internal symmetries or
fine-tuned coupling. These flat bands display remarkable strongly-interacting
phases of matter. Originally considered as a theoretical convenience useful for
obtaining exact analytical solutions of ferromagnetism, flat bands have now
been observed in a variety of settings, ranging from electronic systems to
ultracold atomic gases and photonic devices. Here we review the design and
implementation of flat bands and chart future directions of this exciting
field.
