  We consider generalized Orlicz-Morrey spaces $M_{\Phi,\varphi}(\Rn)$
including their weak versions $WM_{\Phi,\varphi}(\Rn)$. In these spaces we
prove the boundedness of the Riesz potential from $M_{\Phi,\varphi_1}(\Rn)$ to
$M_{\Psi,\varphi_2}(\Rn)$ and from $M_{\Phi,\varphi_1}(\Rn)$ to
$WM_{\Psi,\varphi_2}(\Rn)$. As applications of those results, the boundedness
of the commutators of the Riesz potential on generalized Orlicz-Morrey space is
also obtained. In all the cases the conditions for the boundedness are given
either in terms of Zygmund-type integral inequalities on
$(\varphi_{1},\varphi_{2})$, which do not assume any assumption on monotonicity
of $\varphi_{1}(x,r)$, $\varphi_{2}(x,r)$ in r.
