We give a complete characterization of a measure $$\mu $$ governing the boundedness of fractional OperatorFractional integral operator $$\displaystyle J_{\gamma , \mu }f(x)= \int _{X}\, \frac {f(y)}{d(x,y)^{1-\gamma }} d\mu (y), \;\; 0<\gamma <1, $$ defined on a quasi-metric measure space $$(X, d, \mu )$$ from one grand Lebesgue spaces $$L^{p), \theta _1}_{\mu }(X)$$ into another $$L^{q), \theta _2}_{\mu }(X)$$ . Necessary and sufficient conditions on a measure $$\mu $$ guaranteeing the boundedness of the multilinear fractional integral operator $$T_{\gamma , \mu }^{(m)}$$ (defined with respect to a measure $$\mu $$ ) from the product of Lorentz spaces $$\prod _{k=1}^m L^{r_k, s_k}_{\mu }(X)$$ to the Lorentz space $$L^{p,q}_{\mu }(X)$$ are established. From these results we have the similar results for linear fractional integrals $$J_{\gamma , \mu }$$ (i.e., for $$m=1$$ ). Furthermore, we study the generalized fractional integral transform $$T_{\varphi }$$ associated to a measure on a quasi-metric space. We give a characterization of those measures for which these operators are bounded between $$L_p$$ -spaces defined on nonhomogeneous spaces. We also establish necessary and sufficient conditions for the compactness of fractional integral operators from $$L^p_{\mu }(X)$$ to $$L^q_{\mu }(X)$$ with $$1