  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.
