  We argue that supermassive black hole growth in AGN occurs via sequences of
randomly--oriented accretion discs with angular momentum limited by
self--gravity. These stably co-- or counter--align with the black hole spin
with almost equal frequency. Accretion from these discs very rapidly adjusts
the hole's spin parameter to average values $\bar a \sim 0.1-0.3$ (the precise
range depending slightly on the disc vertical viscosity coefficient $\alpha_2$)
from any initial conditions, but with significant fluctuations ($\Delta a\sim
\pm 0.2$) about these. We conclude (a) AGN black holes should on average spin
moderately, with the mean value $\bar a$ decreasing slowly as the mass
increases; (b) SMBH coalescences leave little long--term effect on $\bar a$;
(c) SMBH coalescence products in general have modest recoil velocities, so that
there is little likelihood of their being ejected from the host galaxy; (d)
black holes can grow even from stellar masses to $\sim 5\times 10^9 \msun$ at
high redshift $z\sim 6$; (e) jets produced in successive accretion episodes can
have similar directions, but after several episodes the jet direction deviates
significantly. Rare examples of massive holes with larger spin parameters could
result from prograde coalescences with SMBH of similar mass, and are most
likely to be found in giant ellipticals. We compare these results with
observation.
  (abridged)
