  Frustration in magnetic materials arising from competing exchange
interactions can prevent the system from adopting long-range magnetic order and
can instead lead to a diverse range of novel quantum and topological states
with exotic quasiparticle excitations. Here, we review prominent examples of
such emergent phenomena, including magnetically-disordered and extensively
degenerate spin ices, which feature emergent magnetic monopole excitations,
highly-entangled quantum spin liquids with fractional spinon excitations,
topological order and emergent gauge fields, as well as complex particle-like
topological spin textures known as skyrmions. We provide an overview of recent
advances in the search for magnetically-disordered candidate materials on the
three-dimensional pyrochlore lattice and two-dimensional triangular, kagome and
honeycomb lattices, the latter with bond-dependent Kitaev interactions, and on
lattices supporting topological magnetism. We highlight experimental signatures
of these often elusive phenomena and single out the most suitable experimental
techniques that can be used to detect them. Our review also aims at providing a
comprehensive guide for designing and investigating novel frustrated magnetic
materials, with the potential of addressing some important open questions in
contemporary condensed matter physics.
