SPACE (SPectroscopic All-sky Cosmic Explorer) is a class-M mission proposed to ESA for the Cosmic Vision 2015-2025 call and recently promoted to the next assessment study phase. SPACE will produce the first all-sky spectroscopic survey of the Universe, taking spectra of more than 500 million galaxies over a wide range of redshifts. SPACE will operate in slit mode (MEMS) at R~400 between 0.8 and 1.8micron down to AB~23, providing redshifts to an accuracy of Delta z~0.001, regardless on the presence of bright emission lines, together with the most relevant physical and evolutionary properties. The catalog of spectroscopic redshift will allow to place the ultimate constraints on the Baryon Acoustic Oscillations and the nature of Dark Energy. By obtaining the first 3-D all-sky map of the Universe at z~2 and beyond, SPACE will trace the growth rate of cosmic structures, the large scale structure of luminous baryons and its cosmic evolution. Besides the all-sky survey, SPACE will carry out a deep extragalactic survey over an area of ~10 sq. deg., enabling a most powerful supernova search program, and a galactic plane survey in integral field mode. Approximately 30% of the time will be open. Due to its versatility, SPACE is a ``self-sufficient'' observatory which can attack and solve the most compelling questions on the nature of the Dark Energy without complementary data from the Earth or space. Its unique wide-field capabilities in the near-IR make SPACE the ideal complement to JWST, ALMA, and the future 25-50 m telescopes.