  We prove trace and extension results for fractional Sobolev spaces of order
$s\in(0,1)$. These spaces are used in the study of nonlocal Dirichlet and
Neumann problems on bounded domains. The results are robust in the sense that
the continuity of the trace and extension operators is uniform as $s$
approaches $1$ and our trace spaces converge to $H^{1/2}(\partial \Omega)$. We
apply these results in order to study the convergence of solutions of nonlocal
Neumann problems as the integro-differential operators localize to a symmetric,
second order operator in divergence form.
