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. This map shows 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 (ℓ < 60), the major foreground contaminant is the diffuse thermal dust emission. At small angular scales (ℓ > 500) the clustered cosmic infrared background and residual point sources are the major contaminants. These foregrounds are carefully modelled and subtracted. We thus measure the tSZ power spectrum over angular scales 0.17° ≲ θ ≲ 3.0° that were previously unexplored. The measured tSZ power spectrum is consistent with that expected from the Planck catalogue of SZ sources, with clear evidence of additional signal from unresolved clusters and, potentially, diffuse warm baryons. 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. The measured tSZ power spectrum and high order statistics are used to place constraints on σ8.