The coldest and densest structures of gas and dust in the Universe have unique spectral signatures across the (sub-)millimetre bands ($\nu \approx 30-950$~GHz). The current generation of single dish facilities has given a glimpse of the potential for discovery, while sub-mm interferometers have presented a high resolution view into the finer details of known targets or in small-area deep fields. However, significant advances in our understanding of such cold and dense structures are now hampered by the limited sensitivity and angular resolution of our sub-mm view of the Universe at larger scales. In this context, we present the case for a new transformational astronomical facility in the 2030s, the Atacama Large Aperture Submillimetre Telescope (AtLAST). AtLAST is a concept for a 50-m-class single dish telescope, with a high throughput provided by a 2~deg - diameter Field of View, located on a high, dry site in the Atacama with good atmospheric transmission up to $\nu\sim 1$~THz, and fully powered by renewable energy. We envision AtLAST as a facility operated by an international partnership with a suite of instruments to deliver the transformative science that cannot be achieved with current or in-construction observatories. As an 50m-diameter telescope with a full complement of advanced instrumentation, including highly multiplexed high-resolution spectrometers, continuum cameras and integral field units, AtLAST will have mapping speeds hundreds of times greater than current or planned large aperture ($>$ 12m) facilities. By reaching confusion limits below L$_*$ in the distant Universe, resolving low-mass protostellar cores at the distance of the Galactic Centre, and directly mapping both the cold and the hot (the Sunyaev-Zeldovich effect) circumgalactic medium of galaxies, AtLAST will enable a fundamentally new understanding of the sub-mm Universe.