We present a detailed prescription for how galaxy formation can be modelled in hierarchical theories of structure formation. Our model incorporates the formation and merging of dark matter halos, the shock heating and radiative cooling of baryonic gas gravitationally confined in these halos, the formation of stars regulated by the energy released by evolving stars and supernovae, the merging of galaxies within dark matter halos, and the spectral evolution of the stellar populations that are formed. The procedure that we describe is very flexible and can be applied to any hierarchical clustering theory. We explore the effects of varying the stellar initial mass function, star formation rates and galaxy merging. The results we compare with an extensive range of observational data, including the B and K galaxy luminosity functions, galaxy colours, the Tully-Fisher relation and galaxy number counts.These data strongly constrain the models and enable the relative importance of each of the physical process to be assessed. We present a broadly successful model defined by a plausible choice of parameters. This fiducial model produces a much more acceptable luminosity function than most previous studies. This is achieved through a modest rate of galaxy mergers and strong suppression of star formation in halos of low circular velocity. However, it fails to produce galaxies as red as many observed ellipticals and, compared with the observed Tully-Fisher relation, the model galaxies have circular velocities which are too large. ** uuencoded compressed postscript file containing all text and figures.**