In this paper we study the limit as $p\to \infty$ of minimizers of the fractional $W^{s,p}$-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.