  We review and discuss theoretical studies addressing the possibility of gas
accretion onto black holes occurring at rates exceeding the Eddington limit.
Our focus is on the applications to the growth of black hole seeds at high
redshift. We first present the general notion of Super-Eddington accretion, and
then summarize the different models and numerical simulations developed to
study such regime. We consider optically thick flows in accretion disks as well
as in spherically symmetric envelopes, and devote particular attention to the
widely adopted model based on the SLIM disk solution. While attractive for its
simplicity, the SLIM disk solution is challenged by the latest generation of
three-dimensional radiation (magneto)-hydrodynamical simulations, in which
radiative losses can be an order of magnitude higher, and the mechanisms of
radiation transport is more complex than straight advection as it takes place
in a complex turbulent regime. We then discuss the gas supply rate to the
sub-pc scale accretion disk or envelope from larger scales, revisiting gas
inflow rates in protogalaxies under various conditions. We conclude that in the
dense gaseous nuclei of high-z galaxies the conditions necessary for the onset
of Super Eddington accretion regimes, such as a high optical depth and high gas
supply rates from large scales, should be naturally met. Feedback from the
growing BH seed should not alter significantly such conditions according to the
results of radiation magneto-hydrodynamical simulations of super-critical flows
in accretion disks. Furthermore, based on the required nuclear gas inflow rates
and the tendency of stellar feedback to remove efficiently gas in low mass
halos, we argue that super-critical accretion will be more easily achieved in
relatively sizable halos, with virial masses $M_{\rm vir} > 10^{10}$
M$_{\odot}$, which become more common at $z < 15$.
