  We have undertaken a deep, wide-field HI imaging survey of M31, reaching a
maximum resolution of about 50 pc and 2 km/s across a 95x48 kpc region. The HI
mass and brightness sensitivity at 100 pc resolution for a 25 km/s wide
spectral feature is 1500 M_Sun and 0.28 K. Our study reveals ubiquitous HI
self-opacity features, discernible in the first instance as filamentary local
minima in images of the peak HI brightness temperature. Local minima are
organized into complexes of more than kpc length and are particularly
associated with the leading edge of spiral arm features. Just as in the Galaxy,
there is only patchy correspondence of self-opaque features with CO(1-0)
emission. Localized opacity corrections to the column density exceed an order
of magnitude in many cases and add globally to a 30% increase in the atomic gas
mass over that inferred from the integrated brightness under the usual
assumption of negligible self-opacity. Opaque atomic gas first increases from
20 to 60 K in spin temperature with radius to 12 kpc but then declines again to
20 K beyond 25 kpc. We have extended the resolved star formation law down to
physical scales more than an order of magnitude smaller in area and mass than
has been possible previously. The relation between total-gas-mass- and
star-formation-rate-density is significantly tighter than that with
molecular-mass and is fully consistent in both slope and normalization with the
power law index of 1.56 found in the molecule-dominated disk of M51 at 500 pc
resolution. Below a gas-mass-density of about 5 M_Sun/pc^2, there is a
down-turn in star-formation-rate-density which may represent a real local
threshold for massive star formation at a cloud mass of about 5x10^4 M_Sun.
