The theory of quasi-spherical subsonic accretion onto magnetized
rotating neutron star is reviewed. Different regimes of quasi-spherical accretion onto a neutron star: supersonic (Bondi)
accretion, which takes place when the captured matter cools down rapidly and falls
supersonically towards the neutron-star
magnetosphere, and subsonic (settling) accretion which occurs when
the plasma remains hot until it meets the magnetospheric boundary.
In subsonic accretion, which works at
X-ray luminosities $\lesssim 4\times 10^{36}$~erg~s$^{-1}$, a hot quasi-spherical shell
must form around the magnetosphere, and the
actual accretion rate onto the neutron star is determined by the ability of the plasma to enter the
magnetosphere due to the Rayleigh-Taylor instability. We show how the dimensionless
parameters of the theory can be determined from observations of equilibrium X-ray pulsars
(Vela X-1, GX 301-2).
We also discuss how in the settling accretion theory bright X-ray flares ($\sim 10^{38}-10^{40}$~ergs) observed in supergiant fast X-ray transients (SFXT) may be produced by
sporadic capture of magnetized stellar-wind plasma. At sufficiently low accretion rates,
magnetic reconnection
can enhance the magnetospheric plasma entry rate, resulting in copious production of X-ray photons,
strong Compton cooling and ultimately in unstable
accretion of the entire shell.
A bright flare develops on the free-fall time
scale in the shell, and the typical energy released in an SFXT bright flare corresponds to
the mass of the shell.