For a polynomial $f\in \mathbb Q[x]$, Hilbert's irreducibility theorem asserts that the fiber $f^{-1}(a)$ is irreducible over $\mathbb Q$ for all values $a\in \mathbb Q$ outside a "thin" set of exceptions $R_f$. The problem of describing $R_f$ is closely related to determining the monodromy group of $f$, and has consequences to open image theorems and reducibility in arithmetic dynamics, and to the polynomial version of the question: "can you hear the shape of the drum?". We shall discuss recent progress on describing $R_f$ and its consequences to the above topics.

Based on joint work with Joachim König