Vector fields on manifolds, and in particular, the study of vector fields near isolated singular points (or equilibrium points), has historically played a major role in mathematics. The most basic invariant of a vector field at an isolated singularity is its Poincaré-Hopf index. This innocent looking index appears naturally in many areas of mathematics and can be reinterpreted in the language of differential geometry, intersection theory, commutative algebra, homological algebra, etcetera, and via obstruction theory gives rise to the definition of Chern classes which are among the most important invariants of a manifold.

In this talk I will give a survey of the ideas mentioned above and we will discuss the possibility of generalizing the Poincaré-Hopf index to vector fields on (complex) singular varieties. This is no trivial task since a singular variety does not have a tangent bundle. What should the bundle in question be? What are its characteristic classes? Are these related to the index? In order to answer these questions I will introduce the GSV index, defined by Gómez--Mont, Seade and Verjovsky in 1991. This index gives a very rich and beautiful interplay between the topological, the algebraic and the analytic aspects of the variety in question.

Instead of proving results I will aim to survey ideas; the talk should be pretty much self contained.