We study dark matter halo density profiles in a high-resolution N-body simulation of a ΛCDM cosmology. Our statistical sample contains ∼5000 haloes in the range 1011–1014 h−1 M⊙, and the resolution allows a study of subhaloes inside host haloes. The profiles are parametrized by an NFW form with two parameters, an inner radius rs and a virial radius Rvir, and we define the halo concentration cvir≡Rvirrs. First, we find that, for a given halo mass, the redshift dependence of the median concentration is cvir∝(1+z)−1. This corresponds to rs(z)∼constant, and is contrary to earlier suspicions that cvir does not vary much with redshift. The implications are that high-redshift galaxies are predicted to be more extended and dimmer than expected before. Secondly, we find that the scatter in halo profiles is large, with a 1σΔ(log cvir)=0.18 at a given mass, corresponding to a scatter in maximum rotation velocities of ΔVmaxVmax=0.12. We discuss implications for modelling the Tully–Fisher relation, which has a smaller reported intrinsic scatter. Thirdly, subhaloes and haloes in dense environments tend to be more concentrated than isolated haloes, and show a larger scatter. These results suggest that cvir is an essential parameter for the theory of galaxy modelling, and we briefly discuss implications for the universality of the Tully–Fisher relation, the formation of low surface brightness galaxies, and the origin of the Hubble sequence. We present an improved analytic treatment of halo formation that fits the measured relations between halo parameters and their redshift dependence, and can thus serve semi-analytic studies of galaxy formation.