In this talk I will present a method of derivation of both previously
known and novel integrable finite difference operators using superconformal indices of certain N=1 supersymmetric gauge theories. I will start with a very brief review of the 4d compactifications of various 6d SCFTs leading to the wide class of 4d gauge theories. Then I will show how to derive various elliptic
integrable systems using superconformal indices of these 4d theories as well as some intuition coming from the geometry of 6d compactifications. Finally I will briefly discuss novel method of obtaining eigenfunctions of these integrable operators using our field theory constructions.