The paper presents a study of dark-matter halo density profiles in a high-resolution N-body simulation of a ΛCDM cosmology. The simulation includes approximately 5000 haloes in the mass range 10¹¹–10¹⁴ h⁻¹ M☉, allowing for the study of subhaloes within host haloes. The profiles are parameterized by an NFW form with two parameters: an inner radius r_s and a virial radius R_vir. The halo concentration c_vir is defined as R_vir / r_s. The study finds that for a given halo mass, the redshift dependence of the median concentration is c_vir ∝ (1+z)⁻¹, which corresponds to r_s(z) ∼ constant. This contradicts earlier suspicions that c_vir does not vary much with redshift. The results suggest that high-redshift galaxies are predicted to be more extended and dimmer than expected before. The scatter in halo profiles is large, with a 1σ Δ(log c_vir) = 0.18 at a given mass, corresponding to a scatter in maximum rotation velocities of ΔV_max / V_max = 0.12. The study also finds that subhaloes and haloes in dense environments tend to be more concentrated than isolated haloes, and show a larger scatter. These results suggest that c_vir is an essential parameter for the theory of galaxy modelling. The paper discusses implications for the universality of the Tully-Fisher relation, the formation of low surface brightness galaxies, and the origin of the Hubble sequence. An improved analytic treatment of halo formation is presented that fits the measured relations between halo parameters and their redshift dependence, and can thus serve semi-analytic studies of galaxy formation. The paper also discusses the redshift dependence of halo properties and the toy-model fits. The results show that the concentration of haloes is proportional to (1+z)⁻¹, which is different from the NFW prediction in which the concentration is a much weaker function of redshift. The study also finds that the concentration of haloes is a decreasing function of c_vir, which has implications for the observed rotation curves and the Tully-Fisher relation. The paper concludes that the results suggest that c_vir is an essential parameter for the theory of galaxy modelling.The paper presents a study of dark-matter halo density profiles in a high-resolution N-body simulation of a ΛCDM cosmology. The simulation includes approximately 5000 haloes in the mass range 10¹¹–10¹⁴ h⁻¹ M☉, allowing for the study of subhaloes within host haloes. The profiles are parameterized by an NFW form with two parameters: an inner radius r_s and a virial radius R_vir. The halo concentration c_vir is defined as R_vir / r_s. The study finds that for a given halo mass, the redshift dependence of the median concentration is c_vir ∝ (1+z)⁻¹, which corresponds to r_s(z) ∼ constant. This contradicts earlier suspicions that c_vir does not vary much with redshift. The results suggest that high-redshift galaxies are predicted to be more extended and dimmer than expected before. The scatter in halo profiles is large, with a 1σ Δ(log c_vir) = 0.18 at a given mass, corresponding to a scatter in maximum rotation velocities of ΔV_max / V_max = 0.12. The study also finds that subhaloes and haloes in dense environments tend to be more concentrated than isolated haloes, and show a larger scatter. These results suggest that c_vir is an essential parameter for the theory of galaxy modelling. The paper discusses implications for the universality of the Tully-Fisher relation, the formation of low surface brightness galaxies, and the origin of the Hubble sequence. An improved analytic treatment of halo formation is presented that fits the measured relations between halo parameters and their redshift dependence, and can thus serve semi-analytic studies of galaxy formation. The paper also discusses the redshift dependence of halo properties and the toy-model fits. The results show that the concentration of haloes is proportional to (1+z)⁻¹, which is different from the NFW prediction in which the concentration is a much weaker function of redshift. The study also finds that the concentration of haloes is a decreasing function of c_vir, which has implications for the observed rotation curves and the Tully-Fisher relation. The paper concludes that the results suggest that c_vir is an essential parameter for the theory of galaxy modelling.