We use N-body simulations to study the evolution of cuspy cold dark matter (CDM) haloes in the gravitational potential of a massive host. Tidal mass-losses reshape CDM haloes, leaving behind bound remnants whose characteristic densities are set by the mean density of the host at the pericentre of their respective orbit. The evolution to the final bound remnant state is essentially complete after ∼5 orbits for nearly circular orbits, while reaching the same remnant requires, for the same pericentre, ∼25 and ∼40 orbits for eccentric orbits with 1:5 and 1:20 pericentre-to-apocentre ratios, respectively. The density profile of tidal remnants is fully specified by the fraction of mass lost, and approaches an exponentially truncated Navarro–Frenk–White profile in the case of heavy mass-loss. Resolving tidal remnants requires excellent numerical resolution; poorly resolved subhaloes have systematically lower characteristic densities and are more easily disrupted. Even simulations with excellent spatial and time resolution fail when the final remnant is resolved with fewer than 3000 particles. We derive a simple empirical model that describes the evolution of the mass and the density profile of the tidal remnant applicable to a wide range of orbital eccentricities and pericentric distances. Applied to the Milky Way, our results suggest that 10^8–|$10^{10}\, \mathrm{M_{\odot }}$| haloes accreted |$\sim 10\, \mathrm{Gyr}$| ago on 1:10 orbits with pericentric distance |$\sim 10\, \mathrm{kpc}$| should have been stripped to 0.1–1 per cent of their original mass. This implies that estimates of the survival and structure of such haloes (the possible hosts of ultra-faint Milky Way satellites) based on direct cosmological simulations may be subject to substantial revision.