In this paper the author establishes some partial rigidity results for algebraic group actions on twin buildings and applies them to study isomorphisms of Kac-Moody groups over an arbitrary field of cardinality at least four. In particular, the author obtains a detailed description of automorphisms of Kac-Moody groups. He also shows that the Hausdorff topology on a Kac-Moody group is an invariant of the abstract group structure. Finally, the author proves the nonexistence of cocentral homomorphisms of Kac-Moody groups of indefinite type over infinite fields with finite-dimensional target, which provides a partial solution to the linearity problem for Kac-Moody groups.