  We review numerous arguments for primordial black holes (PBHs) based on
observational evidence from a variety of lensing, dynamical, accretion and
gravitational-wave effects. This represents a shift from the usual emphasis on
PBH constraints and provides what we term a positivist perspective.
Microlensing observations of stars and quasars suggest that PBHs of around
$1\,M_{\odot}$ could provide much of the dark matter in galactic halos, this
being allowed by the Large Magellanic Cloud microlensing observations if the
PBHs have an extended mass function. More generally, providing the mass and
dark matter fraction of the PBHs is large enough, the associated Poisson
fluctuations could generate the first bound objects at a much earlier epoch
than in the standard cosmological scenario. This simultaneously explains the
recent detection of high-redshift dwarf galaxies, puzzling correlations of the
source-subtracted infrared and X-ray cosmic backgrounds, the size and the
mass-to-light ratios of ultra-faint-dwarf galaxies, the dynamical heating of
the Galactic disk, and the binary coalescences observed by LIGO/Virgo/KAGRA in
a mass range not usually associated with stellar remnants. Even if PBHs provide
only a small fraction of the dark matter, they could explain various other
observational conundra, and sufficiently large ones could seed the supermassive
black holes in galactic nuclei or even early galaxies themselves. We argue that
PBHs would naturally have formed around the electroweak, quantum chromodynamics
and electron-positron annihilation epochs, when the sound-speed inevitably
dips. This leads to an extended PBH mass function with a number of distinct
bumps, the most prominent one being at around $1\,M_{\odot}$, and this would
allow PBHs to explain many of the observations in a unified way.
