Abstract: In finite dimensions, a submanifold of Euclidean space is called isoparametric if: (a) its normal bundle is flat; and (b) the shape operators along any parallel normal vector field are conjugate. It follows from theorems of Dadok, Palais-Terng and Thorbergsson that every (complete, connected, irreducible) isoparametric submanifold in Euclidean space of codimension different from two is a principal orbit of the isotropy representation of a symmetric space. In infinite dimensions, one works in the category of proper Fredholm submanifolds in Hilbert space and defines such a submanifold to be isoparametric if it satisfies conditions (a) and (b) above. It follows from a theorem of Heintze and Liu that every (complete, connected, irreducible) isoparametric submanifold in Hilbert space of codimension different from one is homogeneous. On the other hand, Terng has constructed very interesting examples of homogeneous isoparametric submanifolds, principal orbits of the so called P(G,H)-actions, which are essentially isotropy representations of affine Kac-Moody symmetric spaces. In this talk, we will explain our work in progress to show that every every (complete, connected, irreducible) isoparametric submanifold in Hilbert space of codimension different from one is a principal orbit of a P(G,H)-action. Joint work with Ernst Heintze (Augsburg).