Evolution and Final Fates of Rapidly Accreting Supermassive Stars
  The formation of supermassive stars (SMSs) is a possible pathway to seed
supermassive black holes in the early universe. This chapter summarizes recent
theoretical efforts to understand their evolution, highlighting effects of very
rapid accretion at the rates of $\dot{M}_* \gtrsim 0.1~$M$_\odot$ yr$^{-1}$.
Stellar evolution calculations predict that such an accreting SMS has a
characteristic feature, i.e., a very large radius that monotonically increases
as the stellar mass increases. The radius exceeds $7000$ R$_\odot$ (or $30$ AU)
after the star accretes the gas of more than $10^3$ M$_\odot$. We show that the
emergence of the "supergiant protostar" stage is a key for the formation of
SMSs, because resulting radiative feedback against the accretion flow is
substantially weakened during this stage. We also show that these SMSs end
their lives while the accretion continues, and that their final fates vary with
different accretion rates.
