We present six simulations of galactic stellar haloes formed by the tidal disruption of accreted dwarf galaxies in a fully cosmological setting. Our model is based on the Aquarius project, a suite of high-resolution N-body simulations of individual dark matter haloes. We tag subsets of particles in these simulations with stellar populations predicted by the galform semi-analytic model. Our method self-consistently tracks the dynamical evolution and disruption of satellites from high redshift. The luminosity function (LF) and structural properties of surviving satellites, which agree well with observations, suggest that this technique is appropriate. We find that accreted stellar haloes are assembled between 1 < z < 7 from less than five significant progenitors. These progenitors are old, metal-rich satellites with stellar masses similar to the brightest Milky Way dwarf spheroidals (107–108 M⊙). In contrast to previous stellar halo simulations, we find that several of these major contributors survive as self-bound systems to the present day. Both the number of these significant progenitors and their infall times are inherently stochastic. This results in great diversity among our stellar haloes, which amplifies small differences between the formation histories of their dark halo hosts. The masses (∼ 108–109 M⊙) and density/surface-brightness profiles of the stellar haloes (from 10 to 100 kpc) are consistent with expectations from the Milky Way and M31. Each halo has a complex structure, consisting of well-mixed components, tidal streams, shells and other subcomponents. This structure is not adequately described by smooth models. The central regions (<10 kpc) of our haloes are highly prolate (c/a∼ 0.3), although we find one example of a massive accreted thick disc. Metallicity gradients in our haloes are typically significant only where the halo is built from a small number of satellites. We contrast the ages and metallicities of halo stars with surviving satellites, finding broad agreement with recent observations.