In a joint work with Tommaso de Fernex, we prove that in a family of projective threefolds defined over an algebraically closed field, the locus of rational fibers is a countable union of closed subsets of the locus of separably rationally connected fibers. When the ground field has characteristic zero, this implies that the locus of rational fibers in a smooth family of projective threefolds is a countable union of closed subsets of the parameter space. General expectation suggests that the result maybe false in higher dimension.