The natural question of how much smoother integral currents are with respect to their initial definition goes back to the late 1950s and to the origin of the theory with the seminal article of Federer and Fleming. In this seminar I will explain how closely one can approximate an integral current representing a given homology class with a smooth submanifold. This is a joint study with William Browder and Camillo De Lellis, based on some previous preliminary work of the former author together with Frederick Almgren.