ABSTRACT We study the cosmic evolution of the magnetic fields of a large sample of spiral galaxies in a cosmologically representative volume by employing a semi-analytic galaxy formation model and numerical dynamo solver in tandem. We start by deriving time- and radius-dependent galaxy properties using the galform galaxy formation model, which are then fed into the non-linear mean-field dynamo equations. These are solved to give the large-scale (mean) field as a function of time and galactocentric radius for a thin disc, assuming axial symmetry. A simple prescription for the evolution of the small-scale (random) magnetic field component is also adopted. We find that, while most massive galaxies are predicted to have large-scale magnetic fields at redshift $z$ = 0, a significant fraction of them is expected to contain negligible large-scale field. Our model indicates that, for most of the galaxies containing large-scale magnetic fields today, the mean-field dynamo becomes active at $z$ < 3. Moreover, the typical magnetic field strength at any given galactic stellar mass is predicted to decline with time up until the present epoch, in agreement with our earlier results. We compute the radial profiles of pitch angle, and find broad agreement with observational data for nearby galaxies.