Given a group $G$ acting on a set $X$, an element $g$ of $G$ is called a derangement if it acts without fixed points on $X$. The Boston-Shalev conjecture, proved by Fulman and Guralnick, asserts that in a finite simple group $G$ acting transitively on $X$, the proportion of derangements is at least some absolute constant $c>0$. After giving an introduction to the subject, I will present a version of the conjecture for the proportion of   *conjugacy classes* containing derangements in finite groups of Lie type. Much of the proof concerns the "anatomy" of polynomials over finite fields. Joint work with Sean Eberhard.