We use a large sample of isolated dark matter halo pairs drawn from cosmological N-body simulations to identify candidate systems whose kinematics match that of the Local Group of Galaxies (LG). We find, in agreement with the "timing argument" and earlier work, that the separation and approach velocity of the Milky Way (MW) and Andromeda (M31) galaxies favour a total mass for the pair of ∼ 5 × 10 12 M ⊙ . A mass this large, however, is difficult to reconcile with the small relative tangential velocity of the pair, as well as with the small deceleration from the Hubble flow observed for the most distant LG members. Halo pairs that match these three criteria have average masses a factor of ∼ 2 times smaller than suggested by the timing argument, but with large dispersion, spanning more than a decade in mass. Guided by these results, we have selected 12 halo pairs with total mass in the range 1.6-3.6 × 10 12 M ⊙ for the APOSTLE project (A Project Of Simulations of The Local Environment), a suite of resimulations at various numerical resolution levels (reaching up to ∼ 10 4 M ⊙ per gas particle) that use the hydrodynamical code and subgrid physics developed for the EA-GLE project. These simulations reproduce, by construction, the main kinematics of the MW-M31 pair, and produce satellite populations whose overall number, luminosities, and kinematics are in good agreement with observations of the MW and M31 companions. These diagnostics are sensitive to the total mass assumed for the MW-M31 pair; indeed, the LG satellite population would be quite difficult to reproduce for pair masses as high as indicated by the timing argument. The APOSTLE candidate systems thus provide an excellent testbed to confront directly many of the predictions of the ΛCDM cosmology with observations of our local Universe.