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We present an overview of, and first science results from, the Magellanic Edges Survey (MagES), an ongoing spectroscopic survey mapping the kinematics of red clump and red giant branch stars in the highly substructured periphery of the Magellanic Clouds. In conjunction with Gaia astrometry, MagES yields a sample of ~7000 stars with individual 3D velocities that probes larger galactocentric radii than most previous studies. We outline our target selection, observation strategy, data reduction and analysis procedures, and present results for two fields in the northern outskirts ($>10^{\circ}$ on-sky from the centre) of the Large Magellanic Cloud (LMC). One field, located in the vicinity of an arm-like overdensity, displays apparent signatures of perturbation away from an equilibrium disk model. This includes a large radial velocity dispersion in the LMC disk plane, and an asymmetric line-of-sight velocity distribution indicative of motions vertically out of the disk plane for some stars. The second field reveals 3D kinematics consistent with an equilibrium disk, and yields $V_{\text{circ}}=87.7\pm8.0$km s$^{-1}$ at a radial distance of ~10.5kpc from the LMC centre. This leads to an enclosed mass estimate for the LMC at this radius of $(1.8\pm0.3)\times10^{10}\text{M}_{\odot}$.