The distinguishing number of a permutation group G≤\Sym(X) is the smallest number of colours required to colour the points of X such that only the identity of G preserves the colouring. The distinguishing number of a graph, in the traditional sense, is simply the distinguishing number of its automorphism group. Seress proved that every primitive group of degree n other than \Alt(n) and \Sym(n) has distinguishing number 2, except for a short list of known examples (with distinguishing number 3 or 4). In this talk, I will overview previous work on the distinguishing number of groups, before discussing recent joint work with Alice Devillers and Luke Morgan on the distinguishing number of semiprimitive groups. I will highlight the application of our result to graphs.