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 ~1deg^2^ field, including ~17 per cent, which are undetected at K>=25.7mag. We interpret their ultraviolet-to-radio data using MAGPHYS and determine a median redshift of z=2.61+/-0.08 (1{sigma} 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 M_d_=(6.8+/-0.3)x10^8^M_{sun}_ (thus, gas masses of ~10^11^M_{sun}_) and M_*_=(1.26+/-0.05)x10^11^M_{sun}_. 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_*_>=3x10^11^M_{sun}_ 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 M_h_~6x10^12^M_{sun}_, 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.