  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 $\alpha$ and regularity $\epsilon$
introduced by Garc\'{i}a-Cuerva and Gatto as special cases. And the
$(L^{p}(\mu),L^{q}(\mu))$-boundedness for fractional integral operators on
non-homogeneous metric measure spaces is established. From this, the
$(L^{p}(\mu),L^{q}(\mu))$-boundedness for commutators and multilinear
commutators generated by fractional integral operators with $RBMO(\mu)$
function are further obtained. These results in this paper includes the
corresponding results on both the homogeneous spaces and non-doubling measure
spaces.
