  We consider the Dirichlet eigenvalues of the fractional Laplacian
$(-\Delta)^s$, with $s\in (0,1)$, related to a smooth bounded domain $\Omega$.
We prove that there exists an arbitrarily small perturbation
$\tilde\Omega=(I+\psi)(\Omega)$ of the original domain such that all Dirichlet
eigenvalues of the fractional Laplacian associated to $\tilde\Omega$ are
simple. As a consequence we obtain that all Dirichlet eigenvalues of the
fractional Laplacian on an interval are simple. In addition, we prove that for
a generic choice of parameters all the eigenvalues of some non-local operators
are also simple.
