A derangement is a fixed point free permutation. By an old elementary result of Jordan, every transitive finite permutation group contains a derangement. We will discuss some recent refinements of this result on the proportion of derangements together with applications to Brauer groups and rational maps between curves. In particular, we will discuss the Fulman-Guralnick solution of the Boston-Shalev conjecture on a lower bound for the proportion of derangements for finite simple groups.
