  A kagome lattice naturally features Dirac fermions, flat bands and van Hove
singularities in its electronic structure. The Dirac fermions encode topology,
flat bands favour correlated phenomena such as magnetism, and van Hove
singularities can lead to instabilities towards long-range many-body orders,
altogether allowing for the realization and discovery of a series of
topological kagome magnets and superconductors with exotic properties. Recent
progress in exploring kagome materials has revealed rich emergent phenomena
resulting from the quantum interactions between geometry, topology, spin and
correlation. Here we review these key developments in this field, starting from
the fundamental concepts of a kagome lattice, to the realizations of Chern and
Weyl topological magnetism, to various flat-band many-body correlations, and
then to the puzzles of unconventional charge-density waves and
superconductivity. We highlight the connection between theoretical ideas and
experimental observations, and the bond between quantum interactions within
kagome magnets and kagome superconductors, as well as their relation to the
concepts in topological insulators, topological superconductors, Weyl
semimetals and high-temperature superconductors. These developments broadly
bridge topological quantum physics and correlated many-body physics in a wide
range of bulk materials and substantially advance the frontier of topological
quantum matter.
