  A variety X over a field K is of Hilbert type if the set of rational points
X(K) is not thin. We prove that if f: X\to S is a dominant morphism of
K-varieties and both S and all fibers f^{-1}(s), s in S(K), are of Hilbert
type, then so is X. We apply this to answer a question of Serre on products of
varieties and to generalize a result of Colliot-Th'el`ene and Sansuc on
algebraic groups.
