Supersymmetric gauge theories with extended (N=2) supersymmetry are described in the low-energy limit by finite-dimensional integrable systems of Calogero or Toda type. We discuss this phenomenon and its deformations that emerge upon regularization of non-perturbative (instanton) contributions: a one-parametric deformation (Nekrasov-Shatashvili limit of Nekrasov functions) is associated with quantum integrable systems, while a two-parametric deformation can be associated with an additional deformation of the quantum system with Whitham flows, which naturally leads to the complete AGT corrrespondence: the equality of Nekrasov functions and conformal blocks of two-dimensional conformal theories.