We describe the scientific motivations, the mission concept and the instrumentation of SPACE, a class-M mission proposed for concept study at the first call of the ESA Cosmic-Vision 2015-2025 planning cycle. SPACE aims to produce the largest three-dimensional evolutionary map of the Universe over the past 10 billion years by taking near-IR spectra and measuring redshifts for more than half a billion galaxies at 0∈<∈z∈<∈2 down to AB~23 over 3π sr of the sky. In addition, SPACE will also target a smaller sky field, performing a deep spectroscopic survey of millions of galaxies to AB~26 and at 2∈<∈z∈<∈10∈+. These goals are unreachable with ground-based observations due to the ≈500 times higher sky background (see e.g. Aldering, LBNL report num...