Abstract: Submanifolds of Euclidean space with the simplest local invariants are called isoparametric. They form a rich class of interesting submanifolds which are intimately related to compact Lie groups and symmetric spaces. In this talk we also consider infinite dimensional isoparametric submanifolds in Hilbert space which have many properties in common with their finite dimensional counterparts. They turn out to be related to loop groups and affine Kac-Moody groups/algebras.