We use a set of large cosmological N-body simulations to study the internal structure of dark matter haloes which form in scale-free hierarchical clustering models (initial power spectra P(k) ∞ kn with n = 0, −1 and −2) in an Ω = 1 universe. We find that the radius r178 in a halo corresponding to a mean interior overdensity of 178 accurately delineates the quasi-static halo interior from the surrounding infalling material, in agreement with the simple spherical collapse model. The interior velocity dispersion correlates with mass, again in good agreement with the spherical collapse model. Interior to the virial radius r178 the spherically averaged density, circular velocity and velocity dispersion profiles are well fitted by a simple two-parameter analytical model proposed by Navarro, Frenk & White. This model has ρ ∞ r−l at small radii, steepening to ρ ∞ r−3 at large radii, and fits our haloes to the resolution limit of the simulations. The two model parameters, scalelength and mass, are tightly correlated. Lower mass haloes are more centrally concentrated, and so have scalelengths which are a smaller fraction of their virial radius than those of their higher mass counterparts. This reflects the earlier formation times of low-mass haloes. The haloes are moderately aspherical, with typical axial ratios 1 : 0.8 : 0.65 at their virial radii, becoming gradually more spherical towards their centres. The haloes are generically triaxial, but with a slight preference for prolate over oblate configurations, at least for n = −1 and 0. These shapes are maintained by an anisotropic velocity dispersion tensor. The median value of the spin parameter is λ ≈ 0.04, with a weak trend for lower λ at higher halo mass. We also investigate how the halo properties depend on the algorithm used to identify them in the simulations, using both friends-of-friends and spherical overdensity methods. We find that, for groups selected at mean overdensities ∼ 100 – 400 by either method, the properties are insensitive to how the haloes are selected, if the halo centre is taken as the position of the most bound particle.