  This paper describes the processing applied to the Planck High Frequency
Instrument (HFI) cleaned, time-ordered information to produce photometrically
calibrated maps in temperature and (for the first time) in polarization. The
data from the entire 2.5 year HFI mission include almost five independent
full-sky surveys. HFI observes the sky over a broad range of frequencies, from
100 to 857 GHz. To obtain the best accuracy on the calibration over such a
large range, two different photometric calibration schemes have been used. The
545 and 857 GHz data are calibrated using models of planetary atmospheric
emission. The lower frequencies (from 100 to 353 GHz) are calibrated using the
time-variable cosmological microwave background dipole, which we call the
"orbital dipole". This source of calibration only depends on the satellite
velocity with respect to the solar system. Using a CMB temperature of 2.7255
+/- 0.0006 K, it permits an independent measurement of the amplitude of the CMB
solar dipole (3364.3 +/- 1.5 \mu K) which is approximatively 1\sigma\ higher
than the WMAP measurement with a direction that is consistent between both
experiments. We describe the pipeline used to produce the maps of intensity and
linear polarization from the HFI timelines, and the scheme used to set the zero
level of the maps a posteriori. We also summarize the noise characteristics of
the HFI maps in the 2015 Planck data release and present some null tests to
assess their quality. Finally, we discuss the major systematic effects and in
particular the leakage induced by flux mismatch between the detectors that
leads to spurious polarization signal.
