Abstract: Let L–>S be a line bundle on a nonsingular projective surface. I will discuss recent progress concerning the formula conjectured by Lehn in 1999 for the top Segre class of the tautological bundle L^[n] on Hilb(S,n) and the parallel question for vector bundles V—>S. Results of Voisin play a crucial role. The talk represents joint work with A. Marian and D. Oprea.