Langlands correspondence for Hitchin systems -- global and local

Let X be a smooth projective connected complex curve and G be a complex reductive group. The global Langlands correspondence for Hitchin systems predicts an equivalence between  the derived category of quasi-coherent sheaves on the moduli stack of G-Higgs bundles on X, and the similar category for the Langlands dual group of G. After defining the relevant notions and surveying the known results in this direction, I will formulate a local version of this conjecture, where the curve is replaced by a formal disc. I will briefly explain the compatibility of local and global conjectures and present the current state of knowledge.