We discuss two different approaches to integrability within the framework of N=2 supersymmetric gauge theories. First we look at theories with finite number of massive vacua and study their parameter spaces. This approach leads to a description of quantum many-body systems like spin chains. Second, moduli spaces of vacua of N=2 theories can be conveniently described by VEVs of BPS line operators. In this case the dual many-body system counterparts are models of Hitchin type which arise as classical systems. In the talk I will explain how one can relate integrable systems of the above two kinds and derive various dualities between them.