  We give a brief account of the key properties of elliptic hypergeometric
integrals -- a relatively recently discovered top class of transcendental
special functions of hypergeometric type. In particular, we describe an
elliptic generalization of Euler's and Selberg's beta integrals, elliptic
analogue of the Euler-Gauss hypergeometric function and some multivariable
elliptic hypergeometric functions on root systems. The elliptic Fourier
transformation and corresponding integral Bailey lemma technique is outlined
together with a connection to the star-triangle relation and Coxeter relations
for a permutation group. We review also the interpretation of elliptic
hypergeometric integrals as superconformal indices of four dimensional
supersymmetric quantum field theories and corresponding applications to Seiberg
type dualities.
