It is an easy exercise that any transitive permutation group \(G\) on a set \(\Omega\), \(|\Omega|\geq 2\), contains an element, which acts fixed point freely. These elements are called derangements. A much deeper result, using the classification of the finite simple groups, due to \textit{B. Fein}, \textit{W. M. Kantor} and \textit{M. Schacher} [J. Reine Angew. Math. 328, 39-57 (1981; Zbl 0457.13004)] says that there is a derangement of prime power order. In this paper the authors deal with primitive groups \(G\) such that for some \(r\), which divides \(|\Omega|\), there are no derangements of order \(r\). Using the O'Nan-Scott Theorem they reduce this to a question about automorphism groups of simple groups. They prove: Let \(G\) be a primitive group on \(\Omega\), \(F^*(G)\) alternating or sporadic. For some power \(r\) which divides \(|\Omega|\) either there is a derangement of order \(r\) or there is a list of well-defined exceptions. The list is too long to be given here.