We investigate the origins of galaxy morphology (defined by bulge-to-total K-band luminosity) in the Λ cold dark matter (ΛCDM) cosmology using two galaxy formation models based on the Millennium Simulation, one by Bower et al. (the Durham model) and the other by De Lucia & Blaizot [the Max Planck Institut für Astrophysik (MPA) model]. Both models have had considerable success in reproducing a number of observed properties of the local and high-redshift Universe, including star formation rates, the stellar mass function and the luminosity function out to z∼ 5. There are many similarities, but also fundamental disagreements in the predictions of the two models for galaxy morphology. For example, taking into account uncertainties in the available observational data, both produce a realistic morphological mix today, but its evolution is very different. A main cause of this and other differences is the treatment of disc instabilities which play a more prominent role in the Durham model. Our analysis confirms previous theoretical predictions that elliptical galaxies form most of their stars before the bulk of the galaxy are assembled. Spirals tend to have later ‘assembly’ times than ellipticals as a consequence of in-situ star formation. With the exception of the brightest ellipticals (stellar mass M*≳ 2.5 × 1011 h−1 M⊙), we find that major mergers are not the primary mechanism by which most spheroids (ellipticals and spiral bulges) assemble their mass. In fact, the majority of ellipticals (and the overwhelming majority of spirals) never experience a major merger (above the resolution limit of our simulation). Most ellipticals and spiral bulges acquire their stellar mass through minor mergers or disc instabilities. These conclusions are common to both the MPA and Durham models. The rotation properties of spheroids may help to constrain the importance of disc instabilities in these models.