The intersection cohomology of a complex projective variety $X$ agrees with the usual cohomology if $X$ is smooth and satisfies Poincare duality even if $X$ is singular. It has been proven in various contexts (and conjectured in more) that the intersection cohomology may be represented by the $L^2$- cohomology of a Kähler metric defined on the smooth locus of $X$. The various proofs, though different, often depend on a notion of weight which manifests itself either through representation theory, Hodge theory, or metrical decay. In this talk we discuss the relations between these notions of weight and report on new work in this direction.