We propose a new mission called Space Project for Astrophysical and Cosmological Exploration (SPACE) as part of the ESA long term planning Voyage 2050 programme. SPACE will study galaxy evolution at the earliest times, with the key goals of charting the formation of the heavy elements, measuring the evolution of the galaxy luminosity function, tracing the build-up of stellar mass in galaxies over cosmic time, and finding the first super-massive black holes (SMBHs) to form. The mission will exploit a unique region of the parameter space, between the narrow ultra-deep surveys with HST and JWST, and shallow wide-field surveys such as the Roman Space Telescope and EUCLID, and should yield by far the largest sample of any current or planned miss...