A family of sets is said to be intersecting if any two sets in the family have nonempty intersection. Families of sets subject to various intersection conditions have been studied over the last fifty years and a common feature of many of the results in the area is that the extremal families are often quite asymmetric. Motivated by this, Peter Frankl conjectured in 1981 that symmetric intersecting families must be very small; more precisely, Frankl conjectured that a family of subsets of {1, 2,..., n}  where any three sets intersect must have size o(2^n) if its automorphism group is transitive. In this talk, I shall prove this conjecture. Based on joint work with David Ellis.