We predict the formation histories, properties and descendants of Lyman-break galaxies (LBGs) in the Λ cold dark matter cosmology. In our model, which incorporates a top-heavy initial mass function in starbursts, we find that most LBGs are starbursts triggered by minor mergers of galaxies. The duration of the LBG phase is predicted to be quite short, ∼ 20–60 Myr. We investigate the distributions of stellar and halo masses and morphologies for bright (LUV>L*UV) and faint (LUV>01.L*UV) LBGs at z= 3, 6 and 10 [where we classify LBGs according to their rest-frame ultraviolet (UV) luminosities relative the observed characteristic luminosity Graphic at z≈ 3]. Bright LBGs at z= 3 are predicted to have median stellar masses ∼ 1 × 109 h−1 M⊙ and host halo masses ∼ 3 × 1011 h−1 M⊙, and be typically mildly disc dominated in stellar mass. On the other hand, faint LBGs at z= 10 are predicted to have median stellar masses of only ∼ 1 × 107 h−1 M⊙ and host halo masses ∼ 2 × 1010 h−1 M⊙, and be generally bulge dominated. Bright LBGs at z= 3 evolve to present-day galaxies with median stellar mass ∼ 5 × 1010 h−1 M⊙ (comparable to the Milky Way), consisting of roughly equal numbers of disc- and bulge-dominated systems, and hosted by haloes with median mass ∼ 2 × 1013 h−1 M⊙ (corresponding to medium-size galaxy groups). The model predicts that 40 per cent of Milky Way mass galaxies at the present day have a bright LBG progenitor in the redshift range 3