A fixed point free permutation of a set is called a derangement. It is an old result of Jordan that if G is a transitive permutation group on a finite set, then derangements exist. In fact, some anaylsis of this problem predates group theory going back to the early 1700's. Rather remarkably, it was not until 1993 that is proved that the proportion of derangements in G is at least 1/n, where n is the cardinality of the set. We will discuss variations on this theme and applications to division algebras and maps between smooth projective curves over finite fields. The result fails for infinite groups but we will also discuss a result for algebraic groups that shows that aside from some obvious cases, derangements do exist.