  Intermediate-mass black holes (IMBHs) have not been detected beyond any
reasonable doubt through either dynamical or accretion signatures.
Gravitational waves (GWs) represent an unparalleled opportunity to survey the
sky and detect mergers of IMBHs, making it possible for the first time to
constrain their formation, growth, and merger history across cosmic time. While
the current network LIGO-Virgo-KAGRA is significantly limited in detecting
mergers of IMBH binaries, the next generation of ground-based observatories and
space-based missions promise to shed light on the IMBH population through the
detection of several events per year. Here, we asses this possibility by
determining the optimal network of next-generation of GW observatories to
reconstruct the IMBH merger history across cosmic time. We show that Voyager,
the Einstein Telescope, and Cosmic Explorer will be able to constrain the
distribution of the primary masses of merging IMBHs up to $\sim 10^3\ M_\odot$
and with mass ratio $\gtrsim 0.1$, while LISA will complementary do so at
higher mass and smaller mass ratios. Therefore, a network of next-generation
ground-based and space-based observatories will potentially reconstruct the
merger history of IMBHs. Moreover, IMBHs with masses $\lesssim 5\times 10^3\
M_\odot$ could be observed in multiband up to a redshift of $z\approx 4$,
ushering in a new of era GW astronomy.
