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The 3D velocities of M31 and M33 are important for understanding theevolution and cosmological context of the Local Group. Their most massive starsare detected by Gaia, and we use Data Release 2 (DR2) to determine the galaxyproper motions (PMs). We select galaxy members based on, e.g., parallax, PM,color-magnitude-diagram location, and local stellar density. The PM rotation ofboth galaxies is confidently detected, consistent with the known line-of-sightrotation curves: $V_{\rm rot} = -206\pm86$ km s$^{-1}$ (counter-clockwise) forM31, and $V_{\rm rot} = 80\pm52$ km s$^{-1}$ (clockwise) for M33. We measurethe center-of-mass PM of each galaxy relative to surrounding background quasarsin DR2. This yields that $({\mu}_{\alpha*},{\mu}_{\delta})$ equals $(65 \pm 18, -57 \pm 15)$ $\mu$as yr$^{-1}$ for M31, and $(31 \pm 19 , -29 \pm 16)$$\mu$as yr$^{-1}$ for M33. In addition to the listed random errors, eachcomponent has an additional residual systematic error of 16 $\mu$as yr$^{-1}$.These results are consistent at 0.8$\sigma$ and 1.0$\sigma$ with the (2 and 3times higher-accuracy) measurements already available from Hubble SpaceTelescope (HST) optical imaging and VLBA water maser observations,respectively. This lends confidence that all these measurements are robust. Thenew results imply that the M31 orbit towards the Milky Way is somewhat lessradial than previously inferred, $V_{\rm tan, DR2+HST} = 57^{+35}_{-31}$ kms$^{-1}$, and strengthen arguments that M33 may be on its first infall intoM31. The results highlight the future potential of Gaia for PM studies beyondthe Milky Way satellite system.

First Gaia Dynamics of the Andromeda System: DR2 Proper Motions, Orbits, and Rotation of M31 and M33