We investigate the internal structure and density profiles of haloes of mass 1010-1014 M⊙ in the Evolution and Assembly of Galaxies and their Environment (EAGLE) simulations. These follow the formation of galaxies in a Λ cold dark matter Universe and include a treatment of the baryon physics thought to be relevant. The EAGLE simulations reproduce the observed present-day galaxy stellar mass function, as well as many other properties of the galaxy population as a function of time. We find significant differences between the masses of haloes in the EAGLE simulations and in simulations that follow only the dark matter component. Nevertheless, haloes are well described by the Navarro-Frenk-White density profile at radii larger than ˜5 per cent of the virial radius but, closer to the centre, the presence of stars can produce cuspier profiles. Central enhancements in the total mass profile are most important in haloes of mass 1012-1013 M⊙, where the stellar fraction peaks. Over the radial range where they are well resolved, the resulting galaxy rotation curves are in very good agreement with observational data for galaxies with stellar mass M* < 5 × 1010 M⊙. We present an empirical fitting function that describes the total mass profiles and show that its parameters are strongly correlated with halo mass.