Massive black holes are found to be ubiquitous in the nuclei of nearby galaxies, and their growth and evolution appear to be closely linked to that of their hosts. There is also growing evidence that "binary black holes" form when two galaxies collide. In the violently relaxed core of the merged system, they first pair and later bind under the action of gas-dynamical and gravitational torques. When the black holes are a few milliparsec apart, gravitational wave emission is expected to drive their inspiral down to coalescence, in less than a Hubble time. The Laser Interferometer Space Antenna will detect their waves, infer their spin and masses, and trace their assembly. In this seminar, I will describe the sequence of events that lead to the formation of a massive black hole binary following a galaxy-galaxy collision and critically examine a number of open issues on their coalescence and observability.