  We compute and investigate four types of imprint of a stochastic background
of primordial magnetic fields (PMFs) on the cosmic microwave background (CMB)
anisotropies: the impact of PMFs on the CMB spectra; the effect on CMB
polarization induced by Faraday rotation; the impact of PMFs on the ionization
history; magnetically-induced non-Gaussianities; and the magnetically-induced
breaking of statistical isotropy. Overall, Planck data constrain the amplitude
of PMFs to less than a few nanogauss. In particular, individual limits coming
from the analysis of the CMB angular power spectra, using the Planck
likelihood, are $B_{1\,\mathrm{Mpc}}< 4.4$ nG (where $B_{1\,\mathrm{Mpc}}$ is
the comoving field amplitude at a scale of 1 Mpc) at 95% confidence level,
assuming zero helicity, and $B_{1\,\mathrm{Mpc}}< 5.6$ nG for a maximally
helical field.For nearly scale-invariant PMFs we obtain
$B_{1\,\mathrm{Mpc}}<2.0$ nG and $B_{1\,\mathrm{Mpc}}<0.9$ nG if the impact of
PMFs on the ionization history of the Universe is included. From the analysis
of magnetically-induced non-Gaussianity we obtain three different values,
corresponding to three applied methods, all below 5 nG. The constraint from the
magnetically-induced passive-tensor bispectrum is $B_{1\,\mathrm{Mpc}}< 2.8$
nG. A search for preferred directions in the magnetically-induced passive
bispectrum yields $B_{1\,\mathrm{Mpc}}< 4.5$ nG, whereas the the
compensated-scalar bispectrum gives $B_{1\,\mathrm{Mpc}}< 3$ nG. The analysis
of the Faraday rotation of CMB polarization by PMFs uses the Planck power
spectra in $EE$ and $BB$ at 70 GHz and gives $B_{1\,\mathrm{Mpc}}< 1380$ nG. In
our final analysis, we consider the harmonic-space correlations produced by
Alfv\'en waves, finding no significant evidence for the presence of these
waves. Together, these results comprise a comprehensive set of constraints on
possible PMFs with Planck data.
