By introducing variable Fofana's spaces (L^p(·),L^q)^α( ${ℝ}^n$)(1<p(·)<∞, 1≤q,α≤∞) and their pre-dual spaces, we discussed the properties of those spaces. Meanwhile, via applying hierarchical decomposition of function and real variable techniques, the necessary and sufficient conditions for boundedness of fractional integral operators and their commutators on variable Fofana's spaces are obtained.