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We study the future orbital evolution and merging of the MW-M31-M33 system,using a combination of collisionless N-body simulations and semi-analytic orbitintegrations. Monte-Carlo simulations are used to explore the consequences ofvarying the initial phase-space and mass parameters within their observationaluncertainties. The observed M31 transverse velocity implies that the MW and M31will merge t = 5.86 (+1.61-0.72) Gyr from now, after a first pericenter at t =3.87 (+0.42-0.32) Gyr. M31 may (probability p=41%) make a direct hit with theMW (defined here as a first-pericenter distance less than 25 kpc). Most likely,the MW and M31 will merge first, with M33 settling onto an orbit around them.Alternatively, M33 may make a direct hit with the MW first (p=9%), or M33 mayget ejected from the Local Group (p=7%). The MW-M31 merger remnant willresemble an elliptical galaxy. The Sun will most likely (p=85%) end up atlarger radius from the center of the MW-M31 merger remnant than its currentdistance from the MW center, possibly further than 50 kpc (p=10%). The Sun may(p=20%) at some time in the next 10 Gyr find itself moving through M33 (within10 kpc), but while dynamically still bound to the MW-M31 merger remnant. Thearrival and possible collision of M31 (and possibly M33) with the MW is thenext major cosmic event affecting the environment of our Sun and solar systemthat can be predicted with some certainty. (Abridged)

The M31 Velocity Vector. III. Future Milky Way-M31-M33 Orbital Evolution, Merging, and Fate of the Sun