  In this paper, we introduce the fractional anisotropic Orlicz-Sobolev spaces,
and by using some variational methods, we establish the existence or
non-existence of eigenvalues of fractional anisotropic problems involving a
nonlocal integro-differential operator of elliptic type. In each case, the
competition between the growth rates of the anisotropic coefficients plays an
essential role in the description of the set of eigenvalues.
