Author’s abstract: We introduce a notion of a Kähler metric with constant weighted scalar curvature on a compact Kähler manifold \(X\), depending on a fixed real torus \(\mathbb{T}\) in the reduced group of automorphisms of \(X\), and two smooth (weight) functions \(v>0\) and \(w\), defined on the momentum image (with respect to a given Kähler class \(\alpha\) on \(X\)) of \(X\) in the dual Lie algebra of \(\mathbb{T}\). We show that a number of known results obstructing the existence of constant scalar curvature Kähler (cscK) metrics can be extended to the weighted setting. In particular, we introduce a functional \(\mathcal{M}_{v,w}\) on the space of \(\mathbb{T}\)-invariant Kähler metrics in \(\alpha\), extending the Mabuchi energy in the cscK case, and show that if \(\alpha\) is Hodge, then constant weighted scalar curvature metrics in \(\alpha\) are minima of \(\mathcal{M}_{v,w}\). We define a \((v,w)\)-weighted Futaki invariant of a \(\mathbb{T}\)-compatible smooth Kähler test configuration associated to \((X,\alpha,\mathbb{T})\), and show that the boundedness from below of the \((v,w)\)-weighted Mabuchi functional \(\mathcal{M}_{v,w}\) implies a suitable notion of a \((v,w)\)-weighted K-semistability.