ABSTRACT We analyse the physical properties of a large, homogeneously selected sample of ALMA-located sub-millimetre galaxies (SMGs). This survey, AS2UDS, identified 707 SMGs across the ∼1 deg2 field, including ∼17 per cent, which are undetected at K ≳ 25.7 mag. We interpret their ultraviolet-to-radio data using magphys and determine a median redshift of z = 2.61 ± 0.08 (1σ range of z = 1.8–3.4) with just ∼6 per cent at z > 4. Our survey provides a sample of massive dusty galaxies at z ≳ 1, with median dust and stellar masses of Md = (6.8 ± 0.3) × 108 M⊙ (thus, gas masses of ∼1011 M⊙) and M* = (1.26 ± 0.05) × 1011 M⊙. We find no evolution in dust temperature at a constant far-infrared luminosity across z ∼ 1.5–4. The gas mass function of our sample increases to z ∼ 2–3 and then declines at z > 3. The space density and masses of SMGs suggest that almost all galaxies with M* ≳ 3 × 1011 M⊙ have passed through an SMG-like phase. The redshift distribution is well fit by a model combining evolution of the gas fraction in haloes with the growth of halo mass past a critical threshold of Mh ∼ 6 × 1012 M⊙, thus SMGs may represent the highly efficient collapse of gas-rich massive haloes. We show that SMGs are broadly consistent with simple homologous systems in the far-infrared, 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.