  We study the higher regularity of free boundaries in obstacle problems for
integro-differential operators. Our main result establishes that, once free
boundaries are $C^{1,\alpha}$, then they are $C^\infty$. This completes the
study of regular points, initiated in [5]. In order to achieve this, we need to
establish optimal boundary regularity estimates for solutions to linear
nonlocal equations in $C^{k,\alpha}$ domains. These new estimates are the core
of our paper, and extend previously known results by Grubb (for $k=\infty$) and
by the second author and Serra (for $k=1$).
