To take advantage of the astrophysical potential of Gamma-Ray Bursts (GRBs), Chinese and French astrophysicists have engaged the SVOM mission (Space-based multi-band astronomical Variable Objects Monitor). Major advances in GRB studies resulting from the synergy between space and ground observations, the SVOM mission implements space and ground instrumentation. The scientific objectives of the mission put a special emphasis on two categories of GRBs: very distant GRBs at z$>$5 which constitute exceptional cosmological probes, and faint/soft nearby GRBs which allow probing the nature of the progenitors and the physics at work in the explosion. These goals have a major impact on the design of the mission: the on-board hard X-ray imager is sensitive down to 4 keV and computes on line image and rate triggers, and the follow-up telescopes on the ground are sensitive in the NIR. At the beginning of the next decade, SVOM will be the main provider of GRB positions and spectral parameters on very short time scale. The SVOM instruments will operate simultaneously with a wide range of powerful astronomical devices. This rare instrumental conjunction, combined with the relevance of the scientific topics connected with GRB studies, warrants a remarkable scientific return for SVOM. In addition, the SVOM instrumentation, primarily designed for GRB studies, composes a unique multi-wavelength observatory with rapid slew capability that will find multiple applications for the whole astronomy community beyond the specific objectives linked to GRBs. This report lists the scientific themes that will benefit from observations made with SVOM, whether they are specific GRB topics, or more generally all the issues that can take advantage of the multi-wavelength capabilities of SVOM.
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