In this talk he will introduce Hall-Littlewood (HL) polynomials for any finite root system, and outline an efficient method to compute them even when the order of the Weyl group gets relatively large, e.g. for $E_6$. He will also talk about various limits of the superconformal index, a partition function which is essential for probing the structure of quantum field theories with superconformal symmetry, and present a formula for the index for it in which HL polynomials and Weyl characters appear as essential ingredients.  He will finish with some results we have obtained to study large classes of superconformal theories in 4-dimensions that are enabled by the efficient computation of HL polynomials.