  We introduce the concept of $C^{m,\alpha}$-nonlocal operators, extending the
notion of second order elliptic operator in divergence form with
$C^{m,\alpha}$-coefficients. We then derive the nonlocal analogue of the key
existing results for elliptic equations in divergence form, notably the
H\"older continuity of the gradient of the solutions in the case of
$C^{0,\alpha}$-coefficients and the classical Shauder estimates for
$C^{m+1,\alpha}$-coefficients.
  We further apply the regularity results for $C^{m,\alpha}$-nonlocal operators
to derive optimal higher order regularity estimates of Lipschitz graphs with
prescribed Nonlocal Mean Curvature. Applications to nonlocal equation on
manifolds are also provided.
