Hilbert's irreducibility theorem says that for any irreducible polynomial f(X,Y) in two variables over a number field K, there exists x in K such that f(x,Y) is irreducible. This theorem has many applications in number theory, arithmetic geometry and Galois theory, and a field K with this property is called Hilbertian. In this talk I will give an introduction to Hilbert's theorem and explain recent results on the Hilbertianity of certain fields arising from adjoining torsion points of commutative algebraic group, or more generally, from Galois representations. As we look closer into these results, we will find at their core a purely group theoretical criterion for Hilbertianity.