Physicists on the one hand and Ben Zvi, Sakellaridis and Venkatesh on the other hand predict that given a reductive group G acting in Hamiltonian way on some "nice" Poisson variety Y there should be a way to construct some dual Poisson variety endowed with an action of the Langlands dual group. In the talk I will try to do the following:

1) Explain the construction of the dual variety in the case when Y=T*X where X is a smooth affine G-variety

2) Since the construction involves the derived geometric Satake equivalence I will spend some time reviewing what it is

3) Discuss many examples of dual pairs

4) Discuss what to do when Y is not of cotangent type.

5) Discuss certain conjectural (but almost proven) equivalences of categories related to the above constructions due to Ben Zvi, Sakellaridis and Venkatesh and also due to Finkelberg, Ginzburg and Travkin.