Elliptic hypergeometric functions (a survey)

The hierarchy of Barnes multiple zeta and gamma functions indicates the ways for generalization of plain hypergeometric functions in the same hierarchical way. However, the most beautiful hypergeometric identities are not generalized in a straightforward fashion. Their current top extension is reached in association with the elliptic functions. So, we briefly describe elliptic gamma functions, the elliptic beta integral, an elliptic analogue of the Euler-Gauss hypergeometric function and its W(E_7) symmetry, and outline some of their multivariable extensions. Other aspects of these functions to be mentioned include elliptic biorthogonal rational functions, an elliptic extension of the Fourier transform as well as key applications to quantum integrable systems (top solutions of the Yang-Baxter equation) and quantum field theory (superconformal indices and Seiberg dualities).