  An intriguing alternative to cold dark matter (CDM) is that the dark matter
is a light ( $m \sim 10^{-22}$ eV) boson having a de Broglie wavelength
$\lambda \sim 1$ kpc, often called fuzzy dark matter (FDM). We describe the
arguments from particle physics that motivate FDM, review previous work on its
astrophysical signatures, and analyze several unexplored aspects of its
behavior. In particular, (i) FDM halos smaller than about $10^7 (m/10^{-22}
{\rm eV})^{-3/2} M_\odot$ do not form. (ii) FDM halos are comprised of a core
that is a stationary, minimum-energy configuration called a "soliton",
surrounded by an envelope that resembles a CDM halo. (iii) The transition
between soliton and envelope is determined by a relaxation process analogous to
two-body relaxation in gravitating systems, which proceeds as if the halo were
composed of particles with mass $\sim \rho\lambda^3$ where $\rho$ is the halo
density. (iv) Relaxation may have substantial effects on the stellar disk and
bulge in the inner parts of disk galaxies. (v) Relaxation can produce FDM disks
but an FDM disk in the solar neighborhood must have a half-thickness of at
least $300 (m/10^{-22} {\rm eV})^{-2/3}$ pc. (vi) Solitonic FDM sub-halos
evaporate by tunneling through the tidal radius and this limits the minimum
sub-halo mass inside 30 kpc of the Milky Way to roughly $10^8 (m/10^{-22} {\rm
eV})^{-3/2} M_\odot$. (vii) If the dark matter in the Fornax dwarf galaxy is
composed of CDM, most of the globular clusters observed in that galaxy should
have long ago spiraled to its center, and this problem is resolved if the dark
matter is FDM.
