We review the physics and phenomenology of wave dark matter: a bosonic dark matter candidate lighter than about 30 eV. Such particles have a de Broglie wavelength exceeding the average inter-particle separation in a galaxy like the Milky Way, and are well described as classical waves. We outline the particle physics motivations for them, including the QCD axion and ultra-light axion-like-particles such as fuzzy dark matter. The wave nature of the dark matter implies a rich phenomenology: (1) Wave interference leads to order unity density fluctuations on de Broglie scale. A manifestation is vortices where the density vanishes and around which the velocity circulates. There is one vortex ring per de Broglie volume on average. (2) For sufficiently low masses, soliton condensation occurs at centers of halos. The soliton oscillates and random walks, another manifestation of wave interference. The halo/subhalo abundance is suppressed at small masses, but the precise prediction from numerical wave simulations remains to be determined. (3) For ultra-light ~$10^{-22}$ eV dark matter, the wave interference substructures can be probed by tidal streams/gravitational lensing. The signal can be distinguished from that due to subhalos by the dependence on stream orbital radius/image separation. (4) Axion detection experiments are sensitive to interference substructures for moderately light masses. The stochastic nature of the waves affects the interpretation of experiments and motivates the measurement of correlation functions. Current constraints and open questions, covering detection experiments and cosmological/galactic/black-hole observations, are discussed.
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