  We consider overdetermined problems related to the fractional capacity. In
particular we study $s$-harmonic functions defined in unbounded exterior sets
or in bounded annular sets, and having a level set parallel to the boundary. We
first classify the solutions of the overdetermined problems, by proving that
the domain and the solution itself are radially symmetric. Then we prove a
quantitative stability counterpart of the symmetry results: we assume that the
overdetermined condition is slightly perturbed and we measure, in a
quantitative way, how much the domain is close to a symmetric set.
