ABSTRACT We implement a detailed dust model into the L-Galaxies semi-analytical model which includes: injection of dust by type II and type Ia supernovae (SNe) and AGB stars; grain growth in molecular clouds; and destruction due to supernova-induced shocks, star formation, and reheating. Our grain growth model follows the dust content in molecular clouds and the inter-cloud medium separately, and allows growth only on pre-existing dust grains. At early times, this can make a significant difference to the dust growth rate. Above z ∼ 8, type II SNe are the primary source of dust, whereas below z ∼ 8, grain growth in molecular clouds dominates, with the total dust content being dominated by the latter below z ∼ 6. However, the detailed history of galaxy formation is important for determining the dust content of any individual galaxy. We introduce a fit to the dust-to-metal (DTM) ratio as a function of metallicity and age, which can be used to deduce the DTM ratio of galaxies at any redshift. At z ≲ 3, we find a fairly flat mean relation between metallicity and the DTM, and a positive correlation between metallicity and the dust-to-gas (DTG) ratio, in good agreement with the shape and normalization of the observed relations. We also match the normalization of the observed stellar mass–dust mass relation over the redshift range of 0–4, and to the dust mass function at z = 0. Our results are important in interpreting observations on the dust content of galaxies across cosmic time, particularly so at high redshift.