Coalescing, massive black-hole (MBH) binaries are the most powerful sources of gravitational waves (GWs) in the Universe, which makes MBH science a prime focus for ongoing and upcoming GW observatories. The Laser Interferometer Space Antenna (LISA) -- a gigameter scale space-based GW observatory -- will grant us access to an immense cosmological volume, revealing MBHs merging when the first cosmic structures assembled in the Dark Ages. LISA will unveil the yet unknown origin of the first quasars, and detect the teeming population of MBHs of $10^4 - 10^7$ solar masses. forming within protogalactic halos. The Pulsar Timing Array, a galactic-scale GW survey, can access the largest MBHs the Universe, detecting the cosmic GW foreground from inspiraling MBH binaries of about 10^9 solar masses. LISA can measure MBH spins and masses with precision far exceeding that from electromagnetic (EM) probes, and together, both GW observatories will provide the first full census of binary MBHs, and their orbital dynamics, across cosmic time. Detecting the loud gravitational signal of these MBH binaries will also trigger alerts for EM counterpart searches, from decades (PTAs) to hours (LISA) prior to the final merger. By witnessing both the GW and EM signals of MBH mergers, precious information will be gathered about the rich and complex environment in the aftermath of a galaxy collision. The unique GW characterization of MBHs will shed light on the deep link between MBHs of $10^4-10^{10}$ solar masses and the grand design of galaxy assembly, as well as on the complex dynamics that drive MBHs to coalescence.
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