For a polynomial f∈ℚ[x], Hilbert's irreducibility theorem asserts that the fiber f−1(a) is irreducible over ℚ for all values a∈ℚ outside a "thin" set of exceptions Rf. The problem of describing Rf is closely related to determining the monodromy group of f, and has consequences to arithmetic dynamics, the Davenport-Lewis-Schinzel problem, and to the polynomial version of the question: "can you hear the shape of the drum?". We shall discuss recent progress on describing Rf and its consequences to the above topics.   Based on joint work with Joachim König