  We present new limits on the amplitude of potential primordial magnetic
fields (PMFs) using temperature and polarization measurements of the cosmic
microwave background (CMB) from Planck, BICEP2/Keck Array, POLARBEAR, and
SPTpol. We reduce twofold the 95% CL upper limit on the CMB anisotropy power
due to a nearly-scale-invariant PMF, with an allowed B-mode power at
$\ell=1500$ of $D_{\ell=1500}^{BB} < 0.071 \mu K^2$ for Planck versus
$D_{\ell=1500}^{BB} < 0.034 \mu K^2$ for the combined dataset. We also forecast
the expected limits from soon-to-deploy CMB experiments (like SPT-3G, Adv.
ACTpol, or the Simons Array) and the proposed CMB-S4 experiment. Future CMB
experiments should dramatically reduce the current uncertainties, by one order
of magnitude for the near-term experiments and two orders of magnitude for the
CMB-S4 experiment. The constraints from CMB-S4 have the potential to rule out
much of the parameter space for PMFs.
