Newton introduced divided differences to define interpolation polynomials fitting data points. More recently, Demazure and Bernstein-Gelfand-Gelfand have used divided difference operators when studying the cohomology of flag varieties. These operators may be defined more generally on Borel's equivariant cohomology. I shall begin with a brief introduction to equivariant cohomology, discuss the construction of the divided difference operators, and show how they can be used computationally. This talk is based on joint work with Reyer Sjamaar.