We present the first results based on Planck measurements of the CMB temperature and lensing-potential power spectra. The Planck spectra at high multipoles are extremely well described by the standard spatially-flat six-parameter LCDM cosmology. In this model Planck data determine the cosmological parameters to high precision. We find a low value of the Hubble constant, H0=67.3+/-1.2 km/s/Mpc and a high value of the matter density parameter, Omega_m=0.315+/-0.017 (+/-1 sigma errors) in excellent agreement with constraints from baryon acoustic oscillation (BAO) surveys. Including curvature, we find that the Universe is consistent with spatial flatness to percent-level precision using Planck CMB data alone. We present results from an analysis of extensions to the standard cosmology, using astrophysical data sets in addition to Planck and high-resolution CMB data. None of these models are favoured significantly over standard LCDM. The deviation of the scalar spectral index from unity is insensitive to the addition of tensor modes and to changes in the matter content of the Universe. We find a 95% upper limit of r<0.11 on the tensor-to-scalar ratio. There is no evidence for additional neutrino-like relativistic particles. Using BAO and CMB data, we find N_eff=3.30+/-0.27 for the effective number of relativistic degrees of freedom, and an upper limit of 0.23 eV for the summed neutrino mass. Our results are in excellent agreement with big bang nucleosynthesis and the standard value of N_eff=3.046. We find no evidence for dynamical dark energy. Despite the success of the standard LCDM model, this cosmology does not provide a good fit to the CMB power spectrum at low multipoles, as noted previously by the WMAP team. While not of decisive significance, this is an anomaly in an otherwise self-consistent analysis of the Planck temperature data.