In this talk, we will define a notion of Hodge modules with rational singularities. A variety has rational singularities in the usual sense, if it is normal and the Hodge module related to intersection cohomology has rational singularities in the present sense. With this definition of rational singularities for Hodge modules, we will generalize Boutot's theorem. That is, if a reductive group acts on a smooth affine variety with a stable point, and M is an equivariant Hodge module with rational singularities, then the induced Hodge module on the GIT quotient also has rational singularities. This is based on joint work with D. Arapura.