Bound states in the continuum (BICs) are waves that remain localized even though they coexist with a continuous spectrum of radiating waves that can carry energy away. Their very existence defies conventional wisdom. Although BICs were first proposed in quantum mechanics, they are a general wave phenomenon and have since been identified in electromagnetic waves, acoustic waves in air, water waves and elastic waves in solids. These states have been studied in a wide range of material systems, such as piezoelectric materials, dielectric photonic crystals, optical waveguides and fibres, quantum dots, graphene and topological insulators. In this Review, we describe recent developments in this field with an emphasis on the physical mechanisms that lead to BICs across seemingly very different materials and types of waves. We also discuss experimental realizations, existing applications and directions for future work.National Science Foundation (U.S.) (Grants DMR-1307632)Massachusetts Institute of Technology. Institute for Soldier Nanotechnologies (Contract W911NF-13-D- 0001)United States. Department of Energy. Office of Science. Solid-State Solar Thermal Energy Conversion Center (Grant DE-SC0001299)United States-Israel Binational Science Foundation (Award 2013508
