We illustrate the extraordinary discovery potential for extragalactic astrophysics of a far-infrared/submillimetre (far-IR/submm) all-sky spectroscopic survey with a 3-m-class space telescope. Spectroscopy provides a three-dimensional view of the Universe and allows us to take full advantage of the sensitivity of present-day instrumentation, close to fundamental limits, overcoming the spatial confusion that affects broadband far-IR/submm surveys. A space telescope of the 3-m class (which has already been described in recent papers) will detect emission lines powered by star formation in galaxies out to
. It will specifically provide measurements of spectroscopic redshifts, star-formation rates (SFRs), dust masses, and metal content for millions of galaxies at the peak epoch of cosmic star formation and of hundreds of them at the epoch of reionisation. Many of these star-forming galaxies will be strongly lensed; the brightness amplification and stretching of their sizes will make it possible to investigate (by means of follow-up observations with high-resolution instruments like ALMA, JWST, and SKA) their internal structure and dynamics on the scales of giant molecular clouds (40–100 pc). This will provide direct information on the physics driving the evolution of star-forming galaxies. Furthermore, the arcmin resolution of the telescope at submm wavelengths is ideal for detecting the cores of galaxy proto-clusters, out to the epoch of reionisation. Due to the integrated emission of member galaxies, such objects (as well as strongly lensed sources) will dominate at the highest apparent far-IR luminosities. Tens of millions of these galaxy-clusters-in-formation will be detected at
$z \simeq 2 - 3$
–3, with a tail extending out to
, and thousands of detections at
. Their study will allow us to track the growth of the most massive halos well beyond what is possible with classical cluster surveys (mostly limited to
$z\,\lesssim\, 1.5 - 2$
–2), tracing the history of star formation in dense environments and teaching us how star formation and galaxy-cluster formation are related across all epochs. The obscured cosmic SFR density of the Universe will thereby be constrained. Such a survey will overcome the current lack of spectroscopic redshifts of dusty star-forming galaxies and galaxy proto-clusters, representing a quantum leap in far-IR/submm extragalactic astrophysics.