Seiberg duality and superconformal indices

Seiberg duality (suggested in 1994) - this is an "electric-magnetic" duality of two or several four-dimensional N=1 supersymmetric quantum field theories with non-abelian gauge fields, which are conjecturally equivalent to each other in the strong coupling limit (in the superconformal critical point). Superconformal indices were suggested in 2005 as generating functions enumerating BPS states for such field theories. It appeared that these indices coincide with the elliptic hypergeometric integrals constructed by the speaker in 2000. Equalities of such indices, which are established using the "hypergeometric" methods, prove the equivalence of dual theories in the BPS states sector. The talk will be a survey of these facts.