  We introduce a notion of a K\"ahler metric with constant weighted scalar
curvature on a compact K\"ahler manifold $X$, depending on a fixed real torus
$\mathbb{T}$ in the reduced group of automorphisms of $X$, and two smooth
(weight) functions $\mathrm{v}>0$ and $\mathrm{w}$, defined on the momentum
image (with respect to a given K\"ahler class $\alpha$ on $X$) of $X$ in the
dual Lie algebra of $\mathbb{T}$. A number of natural problems in K\"ahler
geometry, such as the existence of extremal K\"ahler metrics and conformally
K\"ahler, Einstein--Maxwell metrics, or prescribing the scalar curvature on a
compact toric manifold reduce to the search of K\"ahler metrics with constant
weighted scalar curvature in a given K\"ahler class $\alpha$, for special
choices of the weight functions $\mathrm{v}$ and $\mathrm{w}$.
  We show that a number of known results obstructing the existence of constant
scalar curvature K\"ahler (cscK) metrics can be extended to the weighted
setting. In particular, we introduce a functional $\mathcal M_{\mathrm{v},
\mathrm{w}}$ on the space of $\mathbb{T}$-invariant K\"ahler metrics in
$\alpha$, extending the Mabuchi energy in the cscK case, and show (following
the arguments of Li and Sano--Tipler in the cscK and extremal cases) that if
$\alpha$ is Hodge, then constant weighted scalar curvature metrics in $\alpha$
are minima of $\mathcal M_{\mathrm{v},\mathrm{w}}$. Motivated by the recent
work of Dervan--Ross and Dyrefelt in the cscK and extremal cases, we define a
$(\mathrm{v},\mathrm{w})$-weighted Futaki invariant of a
$\mathbb{T}$-compatible smooth K\"ahler test configuration associated to $(X,
\alpha, \mathbb{T})$, and show that the boundedness from below of the
$(\mathrm{v},\mathrm{w})$-weighted Mabuchi functional $\mathcal M_{\mathrm{v},
\mathrm{w}}$ implies a suitable notion of a $(\mathrm{v},\mathrm{w})$-weighted
K-semistability.
