The dimensional reduction of general relativity to two spacetime dimensions is an integrable sigma model. This sigma model has an infinite dimensional hidden symmetry called the Geroch group. Under the action of the Geroch group, the Minkowski metric can be mapped to any vacuum metric with two commuting Killing vectors. I will discuss the role of the Geroch group in general relativity and list some open problems. I will describe the relationship of the sigma model to a recently introduced Chern-Simons action on twistor space, the dimensional reduction of supergravity to two spacetime dimensions, and an enhanced version of the Geroch group that operates in one spacetime dimension. Based on 2011.05831, 2106.13252, 2109.00597, 21xx.xxxxx (to appear).