I will describe how ordinary geometry can be extended in various ways, so that the original tangent space of a manifold is extended to an integrable highest weight module of an arbitrary Kac-Moody algebra g and the diffeomorphisms are generalised accordingly. These extensions include as special cases exceptional geometry, where g=E_d is the U-duality algebra in 11-d dimensions and the generalised diffeomorphisms unify the ordinary diffeomorphisms of the d-dimensional internal (compactified) manifold with the gauge transformations of the (11-d)-dimensional maximal supergravity theory. In particular I will describe how the extended geometry can be constructed by extending the Lie algebra g to a Lie superalgebra. The talk is based on the papers 1711.07694 (with M. Cederwall) and 1507.08828.