We use a series of cosmological N-body simulations for a flat Λ cold dark matter (ΛCDM) cosmology to investigate the structural properties of dark matter haloes, at redshift zero, in the mass range 3 × 109 h−1≲Mvir≲ 3 × 1013 h−1 M⊙. These properties include the concentration parameter, c, the spin parameter, λ, and the mean axis ratio, . For the concentration–mass relation we find c∝M−0.11vir in agreement with the model proposed by Bullock et al., but inconsistent with the alternative model of Eke et al. The normalization of the concentration–mass relation, however, is 15 per cent lower than suggested by Bullock et al. The results for λ and are in good agreement with previous studies, when extrapolated to the lower halo masses probed here, while c and λ are anticorrelated, in that high-spin haloes have, on average, lower concentrations. In an attempt to remove unrelaxed haloes from the sample, we compute for each halo the offset parameter, xoff, defined as the distance between the most bound particle and the centre of mass, in units of the virial radius. Removing haloes with large xoff increases the mean concentration by ∼10 per cent, lowers the mean spin parameter by ∼15 per cent, and removes the most prolate haloes. In addition, it largely removes the anticorrelation between c and λ, though not entirely. We also investigate the relation between halo properties and their large-scale environment density. For low-mass haloes we find that more concentrated haloes live in denser environments than their less concentrated counterparts of the same mass, consistent with recent correlation function analyses. Note, however, that the trend is weak compared to the scatter. For the halo spin parameters we find no environment dependence, while there is a weak indication that the most spherical haloes reside in slightly denser environments. Finally, using a simple model for disc galaxy formation we show that haloes that host low surface brightness galaxies are expected to be hosted by a biased subset of haloes. Not only do these haloes have spin parameters that are larger than average, they also have concentration parameters that are ∼15 per cent lower than the average at a given halo mass. We discuss the implications of all these findings for the claimed disagreement between halo concentrations inferred from low surface brightness rotation curves, and those expected for a ΛCDM cosmology.