  We review the history of the self-gravitating Fermi gas in Newtonian gravity
and general relativity. We mention applications to white dwarfs, neutron stars
and dark matter halos. We describe the nature of instabilities and phase
transitions in the self-gravitating Fermi gas as energy (microcanonical
ensemble) or temperature (canonical ensemble) is reduced. When $N<N_{\rm OV}$,
where $N_{\rm OV}$ is the Oppenheimer-Volkoff critical particle number, the
self-gravitating Fermi gas experiences a gravothermal catastrophe at $E_c$
stopped by quantum mechanics (Pauli's exclusion principle). The equilibrium
state has a core-halo structure made of a quantum core (degenerate fermion
ball) surrounded by a classical isothermal halo. When $N>N_{\rm OV}$, a new
turning point appears at an energy $E"_c$ below which the system experiences a
gravitational collapse towards a black hole [P.H. Chavanis, G. Alberti, Phys.
Lett. B 801, 135155 (2020)]. When $N_{\rm OV}<N<N'_*$, the self-gravitating
Fermi gas experiences a gravothermal catastrophe at $E_c$ leading to a fermion
ball, then a gravitational collapse at $E''_c$ leading to a black hole. When
$N>N'_*$, the condensed branch disappears and the instability at $E_c$ directly
leads to a black hole. We discuss implications of these results for dark matter
halos made of massive neutrinos.
