  In this article we consider certain well-known polynomials associated with
graphs including the independence polynomial and the chromatic polynomial.
These polynomials count certain objects in graphs: independent sets in the case
of the independence polynomial and proper colourings in the case of the
chromatic polynomial. They also have interpretations as partition functions in
statistical physics.
  The algorithmic problem of (approximately) computing these types of
polynomials has been studied for close to 50 years, especially using Markov
chain techniques. Around eight years ago, Barvinok devised a new algorithmic
approach based on Taylor's theorem for computing the permanent of certain
matrices, and the approach has been applied to various graph polynomials since
then. This article is intended as a gentle introduction to the approach as well
as a partial survey of associated techniques and results.
