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 by a smooth submanifold. Part of what will be discussed is derived from a joint study with William Browder and Camillo De Lellis, and builds on earlier preliminary work by the former author together with Frederick Almgren.