  Ultralight bosonic fields are compelling dark-matter candidates and arise in
a variety of beyond-Standard-Model scenarios. These fields can tap energy and
angular momentum from spinning black holes through superradiant instabilities,
during which a macroscopic bosonic condensate develops around the black hole.
Striking features of this phenomenon include gaps in the spin-mass distribution
of astrophysical black holes and a continuous gravitational-wave~(GW) signal
emitted by the condensate. So far these processes have been studied in great
detail for scalar fields and, more recently, for vector fields. Here we take an
important step forward in the black-hole superradiance program by computing,
analytically, the instability time scale, the direct GW emission, and the
stochastic background, in the case of massive tensor (i.e., spin-$2$) fields.
Our analysis is valid for any black hole spin and for small boson masses. The
instability of massive spin-$2$ fields shares some properties with the scalar
and vector cases, but its phenomenology is much richer, for example there exist
multiple modes with comparable instability time scales, and the dominant GW
signal is hexadecapolar rather than quadrupolar. Electromagnetic and GW
observations of spinning black holes in the mass range $M\in(1,10^{10})M_\odot$
can constrain the mass of a putative spin-$2$ field in the range $10^{-22}
\lesssim m_b\,{\rm c^2/eV} \lesssim 10^{-10}$. For $10^{-17}\lesssim m_b\,{\rm
c^2/eV}\lesssim 10^{-15}$, the space mission LISA could detect the continuous
GW signal for sources at redshift $z=20$, or even larger.
