We describe the public ESO near-IR variability survey (VVV) scanning the Milky Way bulge and an adjacent section of the mid-plane where star formation activity is high. The survey will take 1929 hours of observations with the 4-metre VISTA telescope during five years (2010-2014), covering ~10^9 point sources across an area of 520 deg^2, including 33 known globular clusters and ~350 open clusters. The final product will be a deep near-IR atlas in five passbands (0.9-2.5 microns) and a catalogue of more than 10^6 variable point sources. Unlike single-epoch surveys that, in most cases, only produce 2-D maps, the VVV variable star survey will enable the construction of a 3-D map of the surveyed region using well-understood distance indicators such as RR Lyrae stars, and Cepheids. It will yield important information on the ages of the populations. The observations will be combined with data from MACHO, OGLE, EROS, VST, Spitzer, HST, Chandra, INTEGRAL, WISE, Fermi LAT, XMM-Newton, GAIA and ALMA for a complete understanding of the variable sources in the inner Milky Way. This public survey will provide data available to the whole community and therefore will enable further studies of the history of the Milky Way, its globular cluster evolution, and the population census of the Galactic Bulge and center, as well as the investigations of the star forming regions in the disk. The combined variable star catalogues will have important implications for theoretical investigations of pulsation properties of stars. ; Comment: accepted for publication in New Astronomy