We extend the seminal work of Sacks and Uhlenbeck into the fractional framework. We look for maps between manifolds M and N that minimize the critical Sobolev W(s,n/s) energy, where n is the dimension of the manifold M in the domain. We prove that in the case when the target manifold N has trivial n-th homotopy group then in every homotopy class of maps between an n-manifold M to N there exists a minimizing fractional harmonic map. In case when the n-th homotopy group of the target manifold is not trivial we prove that there exists a generating set for the n-th homotopy group of the target manifold N consisting of minimizing fractional maps.

We develop new tools which are interesting on their own, such as a removability result for point-singularities and a balanced energy estimate for non-scaling invariant energies.

Joint work with Armin Schikorra.