  In this paper, we study the interplay between Orlicz-Sobolev spaces $L^{M}$
and $W^{1,M}$ and fractional Sobolev spaces $W^{s,p}$. More precisely, we give
some qualitative properties of the new fractional Orlicz-Sobolev space
$W^{s,M}$, where $s\in (0,1)$ and $M$ is an $N-$function. We also study a
related non-local operator, which is a fractional version of the nonhomogeneous
$M$-Laplace operator. As an application, we prove existence of weak solution
for a non-local problem involving the new fractional $M-$Laplacian operator.
