  The flattening of single-particle band structures plays an important role in
the quest for novel quantum states of matter due to the crucial role of
interactions. Recent advances in theory and experiment made it possible to
construct and tune systems with nearly flat bands, ranging from graphene
multilayers and moire' materials to kagome' metals and ruthenates. While
theoretical models predict exactly flat bands under certain ideal conditions,
evidence was provided that these systems host high-order Van Hove points, i.e.,
points of high local band flatness and power-law divergence in energy of the
density of states. In this review, we examine recent developments in
engineering and realising such weakly dispersive bands. We focus on high-order
Van Hove singularities and explore their connection to exactly flat bands. We
provide classification schemes and discuss interaction effects. We also review
experimental evidence for high-order Van Hove singularities and point out
future research directions.
