The purpose of this chapter is to give an elementary introduction to the most fundamental aspects of elliptic hypergeometric functions. We will also give some indication of their historical origin in statistical mechanics. We will explain how elliptic hypergeometric functions can be used to construct biorthogonal rational functions, generalizing the famous Askey scheme of orthogonal polynomials. Apart from some general mathematical maturity, the only prerequisites will be elementary linear algebra and complex function theory.

We give a survey of elliptic hypergeometric functions associated with root systems, comprised of three main parts. The first two in essence form an annotated table of the main evaluation and transformation formulas for elliptic hypergeometric integrals and series on root systems. The third and final part gives an introduction to Rains' elliptic Macdonald-Koornwinder theory (in part also developed by Coskun and Gustafson). We survey the main properties of elliptic BC_n interpolation functions and BC_n-symmetric biorthogonal functions, which generalize Okounkov's BC_n interpolation Macdonald polynomials and the Koornwinder polynomials, respectively.