Loop groups and their central extensions are arguably the most accessible infinite dimensional groups to be used in Hamiltonian geometry. The theory of Hamiltonian actions of loop groups started with pioneering works by Donaldson and Meinrenken-Woodward, and after several decades of development its main elements are now in place.
In this talk, we will recall the main examples of loop group Hamiltonian spaces including (infinite dimensional) moduli of flat connections, and we will review the main results of the theory including the reduction and convexity theorems, Duistermaat-Heckman localization, and (if time permits) Kirwan surjectivity and [Q,R]=0 (quantization commutes with reduction).