Let G be a group acting transitively on a set X. We will mostly deal with the case that G is finite. We are interested in getting lower bounds for the number of derangements (fixed point free permutations) in G. Certainly, there are always derangments. We will discuss some recent joint work with Jason Fulman regarding a conjecture of Aner Shalev. We will also mention some applications to images of rational points for maps on curves (and higher dimensional varieties) over finite fields and to probabilistic group theory.