Spitzer Space Telescope and Herschel Space Observatory imaging of M31 is used, with a physical dust model, to construct maps of dust surface density, dust-to-gas ratio, starlight heating intensity, and PAH abundance, out to R=25kpc. The global dust mass is M_d=5.4x10^7Msol, the global dust/H mass ratio is M_d/M_H=0.0081, and the global PAH abundance isView original: http://arxiv.org/abs/1306.2304
=0.039. The dust surface density has an inner ring at R=5.6kpc, a maximum at R=11.2kpc, and an outer ring at R=15.1kpc. The dust/gas ratio varies from M_d/M_H=0.026 at the center to 0.0027 at R=25kpc. From the dust/gas ratio, we estimate the ISM metallicity to vary by a factor ~10, from Z/Zsol=3 at R=0 to ~0.3 at R=25kpc. The dust heating rate parameter peaks at the center, with approx 35, declining to approx 0.25 at R=20kpc. Within the central kpc, the starlight heating intensity inferred from the dust modeling is close to what is estimated from the stars in the bulge. The PAH abundance reaches a peak q_PAH=0.045 at R=11.2kpc. When allowance is made for the different spectrum of the bulge stars, q_PAH for the dust in the central kpc is similar to the overall value of q_PAH in the disk. The silicate-graphite-PAH dust model used here is generally able to reproduce the observed dust spectral energy distribution across M31, but overpredicts 500um emission at R=2-6kpc, suggesting that at R=2-6kpc, the dust opacity varies more steeply with frequency (with beta approx 2.3 between 200 and 600um) than in the model.