The lunar surface allows a unique way forward in cosmology, to go beyond current limits. The far side provides an unexcelled radio-quiet environment for probing the dark ages via 21 cm interferometry to seek elusive clues on the nature of the infinitesimal fluctuations that seeded galaxy formation. Far-infrared telescopes in cold and dark lunar polar craters will probe back to the first months of the Big Bang and study associated spectral distortions in the CMB. Optical and IR megatelescopes will image the first star clusters in the Universe and seek biosignatures in the atmospheres of unprecedented numbers of nearby habitable zone exoplanets. The goals are compelling and a stable lunar platform will enable construction of telescopes that can access trillions of modes in the sky, providing the key to exploration of our cosmic origins. This article is part of a discussion meeting issue ‘Astronomy from the Moon the next decades’.The lunar surface has been exposed to the space environment for billions of years and during this time has accumulated records of a wide range of astrophysical phenomena. These include solar wind particles and the cosmogenic products of solar particle events which preserve a record of the past evolution of the Sun, and cosmogenic nuclides produced by high-energy galactic cosmic rays which potentially record the galactic environment of the Solar System through time. The lunar surface may also have accreted material from the local interstellar medium, including supernova ejecta and material from interstellar clouds encountered by the Solar System in the past. Owing to the Moon’s relatively low level of geological activity, absence of an atmosphere, and, for much of its history, lack of a magnetic field, the lunar surface is ideally suited to collect these astronomical records. Moreover, the Moon exhibits geological processes able to bury and thus both preserve and ‘time-stamp’ these records, although gaining access to them is likely to require a significant scientific infrastructure on the lunar surface. This article is part of a discussion meeting issue ‘Astronomy from the Moon the next decades’.Following earlier proposals for optical stellar interferometer concepts in space and on the Moon, the improved ‘hypertelescope’ version capable of direct high-resolution imaging with a high limiting magnitude became tested on Earth, proposed for space, and is now also proposed for the Moon. Many small mirrors can be dilutely arrayed in a lunar impact crater spanning 10-25 km. And a larger version, modified for a flat lunar site and spanning up to several hundred kilometres can be built later if needed for a higher resolution and limiting magnitude. Even larger versions, at the scale of many thousand kilometres, also appear feasible in space at some stage, in the form of a controlled flotilla of mirrors. Among the varied science targets considered with the imaging resolution expected, reaching 100 nano-arcseconds on the Moon, are (a) the early detection and resolved imaging of Near Earth Objects, and their monitoring for eventual collision avoidance by orbital deflection; (b) multi-pixel imaging of exoplanets as part of the search for exolife by mapping local seasonal spectral variations; (c) the physics of neutron stars and black holes at the galactic centre and in other Active Galactic Nuclei; and (d) distant galaxies of cosmological interest. This article is part of a discussion meeting issue ‘Astronomy from the Moon the next decades’.A 20 m space telescope is described with an unvignetted 1° field of view-a hundred times larger in area than fields of existing space telescopes. Its diffraction-limited images are a hundred times sharper than from wide-field ground-based telescopes and extend over much if not all the field, 40 arcmin diameter at 500 nm wavelength, for example. The optical system yielding a 1°, 1.36 m diameter image at f/3.9 has relatively small central obscuration, 9% by area on axis, and is fully baffled. Several carousel-mounted instruments can each access directly the full image. The initial instrument complement includes a 400 gigapixel silicon imager with 2 µm pixels (0.005 arcsec), and a 60 gigapixel HgCdTe imager with 5 µm pixels (0.012 arcsec). A multi-object spectrograph with 10 000 fibres will allow spectroscopy with 0.02 arcsec resolution. Selleck DL-Buthionine-Sulfoximine Direct imaging and spectroscopy of exoplanets can take advantage of the un-aberrated, on-axis image (5 nm RMS wavefront error). While this telescope could be built for operation in free space, a site accessible to a human outpost at the Moon’s south pole would be advantageous, for assembly and repairs. The lunar site would allow also for the installation of new instruments to keep up with evolving scientific priorities and advancing technology. Cooling to less than 100E K would be achieved with a surrounding cylindrical thermal shield. This article is part of a discussion meeting issue ‘Astronomy from the Moon the next decades’.The initial conditions for the density perturbations in the early Universe, which dictate the large-scale structure and distribution of galaxies we see today, are set during inflation. Measurements of primordial non-Gaussianity are crucial for distinguishing between different inflationary models. Current measurements of the matter power spectrum from the cosmic microwave background only constrain this on scales up to k ∼ 0.1 Mpc-1. Reaching smaller angular scales (higher values of k) can provide new constraints on non-Gaussianity. A powerful way to do this is by measuring the HI matter power spectrum at [Formula see text]. In this paper, we investigate what values of k can be reached for the Low-Frequency Array (LOFAR), which can achieve [Formula see text]1″ resolution at approximately 50 MHz. Combining this with a technique to isolate the spectrally smooth foregrounds to a wedge in [Formula see text]-k⊥ space, we demonstrate what values of k we can feasibly reach within observational constraints. We find that LOFAR is approximately five orders of magnitude away from the desired sensitivity, for 10 years of integration time. This article is part of a discussion meeting issue ‘Astronomy from the Moon the next decades’.