Given a polynomial f in Q[X], the Hilbert–Siegel problem asks to describe the set Red(f) of integers a such that f(X)-a is reducible over Q. In this talk, I will present recent joint work with Joachim König and Danny Neftin in which we almost solve this problem. Our approach can be viewed as an application of the Davenport–Lewis–Schinzel problem on the reducibility of the curves f(x)=g(y) for rational functions g in Q[X]. The key ingredients in our work involve monodromy groups and finite group theory.