  We have constructed the first all-sky map of the thermal Sunyaev-Zeldovich
(tSZ) effect by applying specifically tailored component separation algorithms
to the 100 to 857 GHz frequency channel maps from the Planck survey. These maps
show an obvious galaxy cluster tSZ signal that is well matched with blindly
detected clusters in the Planck SZ catalogue. To characterize the signal in the
tSZ map we have computed its angular power spectrum.
  At large angular scales ($\ell < 60$), the major foreground contaminant is
the diffuse thermal dust emission. At small angular scales ($\ell > 500$) the
clustered Cosmic Infrared Background (CIB) and residual point sources are the
major contaminants.
  These foregrounds are carefully modelled and subtracted.
  We measure the tSZ power spectrum in angular scales, $0.17^{\circ} \lesssim
\theta \lesssim 3.0^{\circ}$, that were previously unexplored. The measured tSZ
power spectrum is consistent with that expected from the Planck catalogue of SZ
sources, with additional clear evidence of signal from unresolved clusters and,
potentially, diffuse warm baryons.
  We use the tSZ power spectrum to obtain the following cosmological
constraints:
  $\sigma_8(\Omega_{\mathrm{m}}/0.28)^{3.2/8.1}=0.784 \pm 0.016 (68% C.L.).
  Marginalized band-powers of the Planck tSZ power spectrum and the best-fit
model are given.
  The non-Gaussianity of the Compton parameter map is further characterized by
computing its 1D probability distribution function and its bispectrum. These
are used to place additional independent constraints on $\sigma_{8}$.
