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
(φ1,φ2), which do not assume any assumption on monotonicity
of φ1(x,r), φ2(x,r) in r.Comment: 23 pages. J. Funct. Spaces Appl.(to appear
