There are 8000 galaxies, including 1600 at z ≥ 1.6, which could be simultaneously observed in an E-ELT field of view of 40 arcmin2. A considerable fraction of astrophysical discoveries require large statistical samples, which can only be obtained with multi-object spectrographs (MOS). MOSAIC will provide a vast discovery space, enabled by a multiplex of 200 and spectral resolving powers of R=5000 and 20000. MOSAIC will also offer the unique capability of more than 10 `high-definition' (multi-object adaptive optics, MOAO) integral-field units, optimised to investigate the physics of the sources of reionization. The combination of these modes will make MOSAIC the world-leading MOS facility, contributing to all fields of contemporary astronomy, from extra-solar planets, to the study of the halo of the Milky Way and its satellites, and from resolved stellar populations in nearby galaxies out to observations of the earliest ‘first-light’ structures in the Universe. It will also study the distribution of the dark and ordinary matter at all scales and epochs of the Universe. Recent studies of critical technical issues such as sky-background subtraction and MOAO have demonstrated that such a MOS is feasible with state-of-the-art technology and techniques. Current studies of the MOSAIC team include further trade-offs on the wavelength coverage, a solution for compensating for the non-telecentric new design of the telescope, and tests of the saturation of skylines especially in the near-IR bands. In the 2020s the E-ELT will become the world's largest optical/IR telescope, and we argue that it has to be equipped as soon as possible with a MOS to provide the most efficient, and likely the best way to follow-up on James Webb Space Telescope (JWST) observations.