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I will discuss a recent result obtained in joint work with Robert Guralnick and Martin Liebeck: If G is a finite primitive non-regular permutation group of degree n, then G contains an element fixing at least one and at most n^{1/3} points. This confirms an old conjecture of Peter Neumann. The case where G is affine was addressed by Guralnick and Malle, and our work focuses on the case where G is not affine.