If massive black holes (BHs) are ubiquitous in galaxies and galaxies experience multiple mergers during their cosmic assembly, then BH binaries should be common, albeit temporary, features of most galactic bulges. Observationally, the paucity of active BH pairs points toward binary lifetimes far shorter than the Hubble time, indicating rapid inspiral of the BHs down to the domain where gravitational waves lead to their coalescence. Here, we review a series of studies on the dynamics of massive BHs in gas-rich galaxy mergers that underscore the vital role played by a cool, gaseous component in promoting the rapid formation of the BH binary. The BH binary is found to reside at the center of a massive self-gravitating nuclear disk resulting from the collision of the two gaseous disks present in the mother galaxies. Hardening by gravitational torques against gas in this grand disk is found to continue down to sub-parsec scales. The eccentricity decreases with time to zero and when the binary is circular, accretion sets in around the two BHs. When this occurs, each BH is endowed with its own small-size (≲ 0.01 pc) accretion disk comprising a few percent of the BH mass. Double AGN activity is expected to occur on an estimated timescale of ≲1 Myr. The double nuclear point-like sources that may appear have typical separation of ≲ 10 pc, and are likely to be embedded in the still ongoing starburst. We note that a potential threat of binary stalling, in a gaseous environment, may come from radiation and/or mechanical energy injections by the BHs. […]