  Our Galaxy and the nearby Andromeda galaxy (M31) are the most massive members
of the Local Group, and they seem to be a bound pair, despite the uncertainties
on the relative motion of the two galaxies. A number of studies have shown that
the two galaxies will likely undergo a close approach in the next 4$-$5 Gyr. We
used direct $N$-body simulations to model this interaction to shed light on the
future of the Milky Way - Andromeda system and for the first time explore the
fate of the two supermassive black holes (SMBHs) that are located at their
centers. We investigated how the uncertainties on the relative motion of the
two galaxies, linked with the initial velocities and the density of the diffuse
environment in which they move, affect the estimate of the time they need to
merge and form ``Milkomeda''. After the galaxy merger, we follow the evolution
of their two SMBHs up to their close pairing and fusion. Upon the fiducial set
of parameters, we find that Milky Way and Andromeda will have their closest
approach in the next 4.3 Gyr and merge over a span of 10 Gyr. Although the time
of the first encounter is consistent with other predictions, we find that the
merger occurs later than previously estimated. We also show that the two SMBHs
will spiral in the inner region of Milkomeda and coalesce in less than 16.6 Myr
after the merger of the two galaxies. Finally, we evaluate the
gravitational-wave emission caused by the inspiral of the SMBHs, and we discuss
the detectability of similar SMBH mergers in the nearby Universe ($z\leq 2$)
through next-generation gravitational-wave detectors.
