In this paper, the fractional integral operator on non-homogeneous metric measure spaces is introduced, which contains the classic fractional integral operator, fractional integral with non-doubling measures and fractional integral with fractional kernel of order $��$ and regularity $��$ introduced by Garc��a-Cuerva and Gatto as special cases. And the $(L^{p}(��),L^{q}(��))$-boundedness for fractional integral operators on non-homogeneous metric measure spaces is established. From this, the $(L^{p}(��),L^{q}(��))$-boundedness for commutators and multilinear commutators generated by fractional integral operators with $RBMO(��)$ function are further obtained. These results in this paper includes the corresponding results on both the homogeneous spaces and non-doubling measure spaces.

26 pages