We discuss a nonlocal fractional problem that serves as a nonlocal counterpart of the classical problem of functions of least gradient. We show how we obtain the existence of minimizers by using their connection to nonlocal minimal sets. Additionally, we discuss the continuity of these minimizers and a weak formulation of the problem. The results presented are obtained in collaboration with S. Dipierro, L. Lombardini, J. Maz ́on and E. Valdinoci.