We present several different statistical methods to determine the transverse velocity vector of M31. The underlying assumptions are that the M31 satellites on average follow the motion of M31 through space, and that the galaxies in the outer parts of the Local Group on average follow the motion of the Local Group barycenter through space. We apply the methods to the line-of-sight velocities of 17 M31 satellites, to the proper motions of the 2 satellites M33 and IC 10, and to the line-of-sight velocities of 5 galaxies near the Local Group turn around radius, respectively. This yields 4 independent but mutually consistent determinations of the heliocentric M31 transverse velocities in the West and North directions, with weighted averages <v_W> = -78+/-41 km/s and <v_N> = -38+/-34 km/s. The Galactocentric tangential velocity of M31 is 42 km/s, with 1-sigma confidence interval V_tan <= 56 km/s. The implied M31-Milky Way orbit is bound if the total Local Group mass M exceeds 1.72^{+0.26}_{-0.25}x10^{12} solar masses. If indeed bound, then the timing argument combined with the known age of the Universe implies that M = 5.58^{+0.85}_{-0.72}x10^{12} solar masses. This is on the high end of the allowed mass range suggested by cosmologically motivated models for the individual structure and dynamics of M31 and the Milky Way, respectively. It is therefore possible that the timing mass is an overestimate of the true mass, especially if one takes into account recent results from the Millennium Simulation that show that there is also a theoretical uncertainty of 41 percent (Gaussian dispersion) in timing mass estimates. The M31 transverse velocity implies that M33 is in a tightly bound orbit around M31. This may have led to some tidal deformation of M33. It will be worthwhile to search for observational evidence of this.
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