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In this talk we will discuss the relation between 4d, 5d supersymmetric gauge theories and quantum integrable systems with elliptic potential (such as Calogero-Moser and Ruijsenaars-Schneider). As we will see, an exact analytic solution to such integrable systems can be constructed from VEVs of gauge theory observables; sometimes however these need to be corrected by appropriate terms, non-perturbative in the Omega background parameters, which can be determined from the properties of the integrable system.