Segre classes are a key ingredient in intersection theory, with applications
to enumerative geometry and to the study of singularities, among others. Work done in
collaboration with Carel Faber several years ago can be interpreted as the computation
of many enumeratively relevant Segre classes. We will recall the context of that work
and present more recent advances in the understanding of Segre classes, particularly
an integral formula computing the Segre class of a subscheme of projective space in
terms of an associated Newton-Okounkov body.