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 [1].

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 [1],[2].

I will review the design and implementation of flat bands, classification schemes, discuss recent results on adding many-body interactions and disorder, and chart future directions of this exciting field.

[1] Artificial flat band systems: from lattice models to experiments. Daniel Leykam, Alexei Andreanov, Sergej Flach. Adv. Phys.: X 3, 1473052 (2018)

[2] Photonic Flat Bands. Daniel Leykam, Sergej Flach. APL PHOTONICS 3, 070901 (2018)