A variety X over a field K is of Hilbert type if X(K) is not thin. We prove that if f: X -> S is a dominant morphism of K-varieties and both S and all fibers f(-1)(s), s is an element of 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-Thelene and Sansuc on algebraic groups.