The author considers fractional multilinear integral operators \[ {\mathcal I}_\alpha \vec{f}(x):= \int_{\left({\mathbb R}^n\right)^m } {f_1(y_1) \cdots f_m(y_m)\over \left(|x-y_1|+\dots+|x-y_m|\right)^{nm-\alpha}} d\vec{y}. \] Sufficient conditions for the two weight inequalities \[ \left(\int_{{\mathbb R}^n} (|{\mathcal I}_\alpha\vec{f}|u)^q dx\right)^{1/q} \leq C \prod_{i=1}^m \left(\int_{{\mathbb R}^n} (|f_i|w_i)^{p_i} dx\right)^{1/p_i} \] of these operators are found. For one weight inequalities (\(u:=\prod_{i=1}^m w_i\)) a necessary and sufficient condition is then obtained as a consequence of the two weight inequalities. Similar results are proved for the multilinear fractional maximal functions. As an application, Poincaré and Sobolev inequalities for products of functions are presented.