Planck data have been used to provide stringent new constraints on cosmic strings and other defects. We describe forecasts of the CMB power spectrum induced by cosmic strings, calculating these from network models and simulations using line-of-sight Boltzmann solvers. We have studied Nambu-Goto cosmic strings, as well as field theory strings for which radiative effects are important, thus spanning the range of theoretical uncertainty in strings models. We have added the angular power spectrum from strings to that for a simple adiabatic model, with the extra fraction defined as $f_{10}$ at multipole $ell=10$. This parameter has been added to the standard six parameter fit using COSMOMC with flat priors. For the Nambu-Goto string model, we have obtained a constraint on the string tension of $Gmu/c^2 < 1.5 x 10^{-7}$ and $f_{10} < 0.015$ at 95% confidence that can be improved to $Gmu/c^2 < 1.3 x 10^{-7}$ and $f_{10} < 0.010$ on inclusion of high-$ell$ CMB data. For the abelian-Higgs field theory model we find, $Gmu_{AH}/c^2 < 3.2 x 10^{-7}$ and $f_{10} < 0.028$. The marginalized likelihoods for $f_{10}$ and in the $f_{10}$--$Omega_b h^2$ plane are also presented. We have also obtained constraints on $f_{10}$ for models with semi-local strings and global textures for which $Gmu/c^2 < 1.1 x 10^{-6}$. We have made complementarity searches for the specific non-Gaussian signatures of cosmic strings, calibrating with all-sky Planck resolution CMB maps generated from networks of post-recombination strings. We have obtained upper limits on the string tension at 95% confidence of $Gmu/c^2 < 8.8 x 10^{-7}$ using modal bispectrum estimation and $Gmu/c^2 < 7.8 x 10^{-7}$ for real space searches with Minkowski functionals. These are conservative upper bounds because only post-recombination string contributions have been included in the non-Gaussian analysis.
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