ABSTRACT The ‘core-cusp’ problem is considered a key challenge to the ΛCDM paradigm. Haloes in dark matter only simulations exhibit ‘cuspy’ profiles, where density continuously increases towards the centre. However, the dark matter profiles of many observed galaxies (particularly in the dwarf regime) deviate strongly from this prediction, with much flatter central regions (‘cores’). We use NewHorizon (NH), a hydrodynamical cosmological simulation, to investigate core formation, using a statistically significant number of galaxies in a cosmological volume. Haloes containing galaxies in the upper (M⋆ ≥ 1010.2 M⊙) and lower (M⋆ ≤ 108 M⊙) ends of the stellar mass distribution contain cusps. However, Haloes containing galaxies with intermediate (108 M⊙ ≤ M⋆ ≤ 1010.2 M⊙) stellar masses are generally cored, with typical halo masses between 1010.2 M⊙ and 1011.5 M⊙. Cores form through supernova-driven gas removal from halo centres, which alters the central gravitational potential, inducing dark matter to migrate to larger radii. While all massive (M⋆ ≥ 109.5 M⊙) galaxies undergo a cored-phase, in some cases cores can be removed and cusps reformed. This happens if a galaxy undergoes sustained star formation at high redshift, which results in stars (which, unlike the gas, cannot be removed by baryonic feedback) dominating the central gravitational potential. After cosmic star formation peaks, the number of cores, and the mass of the Haloes they are formed in, remain constant, indicating that cores are being routinely formed over cosmic time after a threshold halo mass is reached. The existence of cores is, therefore, not in tension with the standard paradigm.