We present the consolidated scientific case for multi-object spectroscopy with the MOSAIC concept on the European ELT. The cases span the full range of ELT science and require either high multiplex or high definition observations to best exploit the excellent sensitivity and wide field-of-view of the telescope. Following scientific prioritisation by the Science Team during the recent Phase A study of the MOSAIC concept, we highlight four key surveys designed for the instrument using detailed simulations of its scientific performance. We discuss future ways to optimise the conceptual design of MOSAIC in Phase B, and illustrate its competitiveness and unique capabilities by comparison with other facilities that will be available in the 2020s.
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