  We present new, tight, constraints on the cosmological background of
gravitational waves (GWs) using the latest measurements of CMB temperature and
polarization anisotropies provided by the Planck, BICEP2 and Keck Array
experiments. These constraints are further improved when the GW contribution
$N^{\rm GW}_{\rm eff}$ to the effective number of relativistic degrees of
freedom $N_{\rm eff}$ is also considered. Parametrizing the tensor spectrum as
a power law with tensor-to-scalar ratio $r$, tilt $n_\mathrm{t}$ and pivot
$0.01\,\mathrm{Mpc}^{-1}$, and assuming a minimum value of $r=0.001$, we find
$r < 0.089$, $n_\mathrm{t} = 1.7^{+2.1}_{-2.0}$ ($95\%\,\mathrm{CL}$, no
$N^{\rm GW}_{\rm eff}$) and $r < 0.082$, $n_\mathrm{t} = -0.05^{+0.58}_{-0.87}$
($95\%\,\mathrm{CL}$, with $N^{\rm GW}_{\rm eff}$). When the recently released
$95\,\mathrm{GHz}$ data from Keck Array are added to the analysis, the
constraints on $r$ are improved to $r < 0.067$ ($95\%\,\mathrm{CL}$, no $N^{\rm
GW}_{\rm eff}$), $r < 0.061$ ($95\%\,\mathrm{CL}$, with $N^{\rm GW}_{\rm
eff}$). We discuss the limits coming from direct detection experiments such as
LIGO-Virgo, pulsar timing (European Pulsar Timing Array) and CMB spectral
distortions (FIRAS). Finally, we show future constraints achievable from a
COrE-like mission: if the tensor-to-scalar ratio is of order $10^{-2}$ and the
inflationary consistency relation $n_\mathrm{t} = -r/8$ holds, COrE will be
able to constrain $n_\mathrm{t}$ with an error of $0.16$ at
$95\%\,\mathrm{CL}$. In the case that lensing $B$-modes can be subtracted to
$10\%$ of their power, a feasible goal for COrE, these limits will be improved
to $0.11$ at $95\%\,\mathrm{CL}$.
