The authors consider the generalized fractional maximal operator \[ M_{\rho}f(x)=\sup_{r>0}\rho(x, r)|B(x, r)|^{-1}\int_{B(x, r)}|f(y)|\,dy, \] where a positive function \(\rho(x, r)\), \(x\in\mathbb R^N\), \(r>0\), is continuous in the variable \(r\) for each \(x\), and there exists a constant \(C_1\ge 1\) such that \(\rho(x, t_1)\le C_1\rho(x, t_2)\) for all \(x\) whenever \(0<t_1<t_2<\infty\); the open ball \(B(x, r)\) is centered at \(x\) with radius \(r\). Functions \(\rho_1(x. r)=r^{\alpha(x)}\), \(\rho_2(x. r)=(\log(e+1/r))^{-\beta(x)}\) are typical examples of \(\rho(x, r)\). The boundedness of \(M_{\rho}\) on Musielak-Orlicz-Morrey spaces is proved. The Sobolev and Trudinger type inequalities for non-doubling Musielak-Orlicz-Morrey spaces are obtained.