  This review addresses the issue of whether there are physically realistic
self-similar solutions in which a primordial black hole is attached to an exact
or asymptotically Friedmann model for an equation of state of the form
$p=(\gamma-1)\rho c^2$. In the positive pressure case ($1 < \gamma < 2$), there
is no such solution when the black hole is attached to an exact Friedmann
background via a sonic point. However, it has been claimed that there is a
one-parameter family of asymptotically Friedmann black hole solutions providing
the ratio of the black hole size to the cosmological horizon size is in a
narrow range above some critical value. There are also "universal" black holes
in which the black hole has an apparent horizon but no event horizon. It turns
out that both these types of solution are only asymptotically {\it
quasi}-Friedmann, because they contain a solid angle deficit at large
distances, but they are not necessarily excluded observationally.
  We also consider the possibility of self-similar black hole solutions in a
universe dominated by a scalar field. If the field is massless, the situation
resembles the stiff fluid case, so any black hole solution is again contrived,
although there may still be universal black hole solutions. The situation is
less clear if the scalar field is rolling down a potential and therefore
massive, as in the quintessence scenario. Although no explicit asymptotically
Friedmann black hole solutions of this kind are known, they are not excluded
and comparison with the $0 < \gamma < 2/3$ perfect fluid case suggests that
they should exist if the black hole is not too large. This implies that a black
hole might grow as fast as the cosmological horizon in a quintessence-dominated
universe in some circumstances, supporting the proposal that accretion onto
primordial black holes may have played a role in the production of the
supermassive black holes in galactic nuclei.
