Let $P = a_m T^m + \sum_{j=0}^{m} a_j (T^{m-j} + T^{m+j})$ be a monic reciprocal polynomial of even degree n, with the $a_j$ drawn independently and uniformly at random from $\{1, 2, \ldots, H\}$ for some fixed integer $H \geq 35$. In joint work with Dimitris Koukoulopoulos, building on prior work of Bary-Soroker—Koukoulopoulos—Kozma, we show that $P$ is irreducible over the rationals, and has large Galois group, with probability tending to 1 as $n$ tends to infinity. This is in line with the long-held belief going back in various forms to Hilbert, Van der Waerden, Odlyzko—Poonen and Konyagin that a typical polynomial is irreducible and has Galois group as large as possible. I will discuss some recent, related developments in the field of probabilistic Galois theory before discussing aspects of the proof, which draws from classical polynomial theory, Fourier analysis, probability theory and group theory.