The Transient High Energy Sky and Early Universe Surveyor (THESEUS) will address multiple fundamental questions in modern cosmology and astrophysics. The primary scientific goals of the mission relate to the Early Universe topic of ESA’s Cosmic Vision, "How did the Universe originate and what is it made of?", and, specifically, the sub-topics: the early Universe, the Universe taking shape, and the evolving violent Universe. It will also have a significant impact on the science priorities "The Gravitational Wave Universe" and "The Hot and Energetic Universe". THESEUS will have unique capabilities to: a) explore the Early Universe (the cosmic dawn and re-ionisation era) by unveiling the Gamma-Ray Burst (GRB) population in the first billion years; and b) perform unprecedented deep monitoring of the soft X-ray transient sky, thus providing fundamental synergy with next-generation gravitational wave and neutrino detectors (multi-messenger astrophysics), as well as with major electromagnetic facilities of the coming decades (ATHENA, E-ELT, SKA, CTA, LSST, etc.).