Elliptic Hypergeometric Functions

Elliptic hypergeometric functions are the most complicated special functions of hypergeometric type. In these introductory lectures I'll try to present the ideas behind their construction and outline some of the applications. The topical content is given below.
Multiple zeta and gamma functions of Barnes and infinite basic products. Finite difference equations of the first order with elliptic coefficients and the elliptic gamma functions. The elliptic beta integral as the top know generalization of the Euler beta integral. An elliptic analogue of the Euler-Gauss hypergeometric function and its W(E_7) symmetry. An elliptic analogue of the Selberg integral. Relation to the representation theory of Lie groups (and supergroups) via the interpretation of elliptic hypergeometric integrals as superconformal indices of four dimensional supersymmetric gauge field theories.