This paper presents an all-sky model of dust emission from the Planck 353, 545, and 857 GHz, and IRAS 100 μm data. Using a modified blackbody fit to the data we present all-sky maps of the dust optical depth, temperature, and spectral index over the 353-3000 GHz range. This model is a good representation of the IRAS and Planck data at 5' between 353 and 3000 GHz (850 and 100 μm). It shows variations of the order of 30% compared with the widely-used model of Finkbeiner, Davis, and Schlegel. The Planck data allow us to estimate the dust temperature uniformly over the whole sky, down to an angular resolution of 5', providing an improved estimate of the dust optical depth compared to previous all-sky dust model, especially in high-contrast molecular regions where the dust temperature varies strongly at small scales in response to dust evolution, extinction, and/or local production of heating photons. An increase of the dust opacity at 353 GHz, τ353/NH, from the diffuse to the denser interstellar medium (ISM) is reported. It is associated with a decrease in the observed dust temperature, Tobs, that could be due at least in part to the increased dust opacity. We also report an excess of dust emission at H i column densities lower than 1020 cm-2 that could be the signature of dust in the warm ionized medium. In the diffuse ISM at high Galactic latitude, we report an anticorrelation between τ353/NH and Tobs while the dust specific luminosity, i.e., the total dust emission integrated over frequency (the radiance) per hydrogen atom, stays about constant, confirming one of the Planck Early Results obtained on selected fields. This effect is compatible with the view that, in the diffuse ISM, Tobs responds to spatial variations of the dust opacity, due to variations of dust properties, in addition to (small) variations of the radiation field strength. The implication is that in the diffuse high-latitude ISM τ353 is not as reliable a tracer of dust column density as we conclude it is in molecular clouds where the correlation of τ353 with dust extinction estimated using colour excess measurements on stars is strong. To estimate Galactic E(B - V) in extragalactic fields at high latitude we develop a new method based on the thermal dust radiance, instead of the dust optical depth, calibrated to E(B - V) using reddening measurements of quasars deduced from Sloan Digital Sky Survey data.