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 polycyclic aromatic hydrocarbon (PAH) abundance, out to R≈ 25 kpc. The global dust mass is M d= 5.4× 10 7 M☉, the global dust/H mass ratio is M d/M H= 0.0081, and the global PAH abundance is< q PAH>= 0.039. The dust surface density has an inner ring at R= 5.6 kpc, a maximum at R= 11.2 kpc, and an outer ring at R≈ 15.1 kpc. The dust/gas ratio varies from M d/M H≈ 0.026 at the center to∼ 0.0027 at R≈ 25 kpc. From the dust/gas ratio, we estimate the interstellar medium metallicity to vary by a factor∼ 10, from Z/Z☉≈ 3 at R= 0 to∼ 0.3 at R= 25 kpc. The dust heating rate parameter< U> peaks at the center, with< U>≈ 35, declining to< U>≈ 0.25 at R= 20 kpc. Within the central kiloparsec, 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≈ 11.2 kpc. When allowance is made for the different spectrum of the bulge stars, q PAH for the dust in the central kiloparsec 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 500 μm emission at R≈ 2–6 kpc, suggesting that at R= 2–6 kpc, the dust opacity varies more steeply with frequency (with β≈ 2.3 between 200 and 600 μm) than in the model.