In this paper it is proved that anisotropic fractional maximal operator $M_{\a,\sigma}$, $0 \le \a < |\sigma|$ is bounded on anisotropic generalized Morrey spaces $M_{p,\varphi,\sigma}$, where $|\sigma|=\sum_{i=1}^n \sigma_i$ is the homogeneous dimension of $\Rn$. We find the conditions on the pair $(\varphi_1,\varphi_2)$ which ensure the Spanne-Guliyev type boundedness of the operator $M_{\a,\sigma}$ from anisotropic generalized Morrey space $M_{p,\varphi_1,\sigma}$ to $M_{q,\varphi_2,\sigma}$, $1