We analyse the physical properties of a large, homogeneously selected sample of ALMA-located sub-millimetre galaxies (SMGs) detected in the SCUBA-2 Cosmology Legacy Survey 850-µm map of the UKIDSS/UDS field. This survey, AS2UDS, identified 707 SMGs across the ∼ 1 deg 2 field, including ∼17 per cent which are undetected in the optical/near-infrared to K 25.7 mag. We interpret the UV-to-radio data of these systems using a physically motivated model, magphys and determine a median photometric redshift of z = 2.61±0.08, with a 68 th percentile range of z = 1.8-3.4, with just ∼ 6 per cent at z > 4. The redshift distribution is well fit by a model combining evolution of the gas fraction in halos with the growth of halo mass past a critical threshold of ∼4×10 12 M , thus SMGs may represent the highly efficient collapse of gasrich massive halos. Our survey provides a sample of the most massive, dusty galaxies at z 1, with median dust and stellar masses of M d = (6.8±0.3) × 10 8 M (thus, gas masses of ∼ 10 11 M ) and M * = (1.26±0.05) × 10 11 M . These galaxies have gas fractions of f gas = 0.41±0.02 with depletion timescales of ∼ 150 Myr. The gas mass function evolution of our sample at high masses is consistent with constraints at lower masses from blind CO-surveys, with an increase to z ∼ 2-3 and then a decline at higher redshifts. The space density and masses of SMGs suggests that almost all galaxies with M * 2 × 10 11 M have passed through an SMG-like phase. We find no evolution in dust temperature at a constant far-infrared luminosity across z ∼ 1.5-4. We exploit dust continuum sizes to show that SMGs appear to behave as simple homologous systems in the far-infrared, having properties consistent with a centrally illuminated starburst. Our study provides strong support for an evolutionary link between the active, gas-rich SMG population at z > 1 and the formation of massive, bulge-dominated galaxies across the history of the Universe.