  Primordial Black Holes (PBH) from peaks in the curvature power spectrum could
constitute today an important fraction of the Dark Matter in the Universe. At
horizon reentry, during the radiation era, order one fluctuations collapse
gravitationally to form black holes and, at the same time, generate a
stochastic background of gravitational waves coming from second order
anisotropic stresses in matter. We study the amplitude and shape of this
background for several phenomenological models of the curvature power spectrum
that can be embedded in waterfall hybrid inflation, axion, domain wall, and
boosts of PBH formation at the QCD transition. For a broad peak or a nearly
scale invariant spectrum, this stochastic background is generically enhanced by
about one order of magnitude, compared to a sharp feature. As a result,
stellar-mass PBH from Gaussian fluctuations with a wide mass distribution are
already in strong tension with the limits from Pulsar Timing Arrays, if they
constitute a non negligible fraction of the Dark Matter. But this result is
mitigated by the uncertainties on the curvature threshold leading to PBH
formation. LISA will have the sensitivity to detect or rule out light PBH down
to $10^{-14} M_{\odot}$. Upcoming runs of LIGO/Virgo and future interferometers
such as the Einstein Telescope will increase the frequency lever arm to
constrain PBH from the QCD transition. Ultimately, the future SKA Pulsar Timing
Arrays could probe the existence of even a single stellar-mass PBH in our
Observable Universe.
