  Motivated by the two-dimensional massive gravity description of
$T\overline{T}$ deformations, we propose a direct generalization in $d$
dimensions. Our methodology indicates that all terms up to order $d$ are
present in the deformation. In two dimensions, $T\overline{T}$ is enhanced by a
linear and a constant term, and exhibits an interesting behaviour regarding the
deformed spectrum and correlators. At certain limits, this deformation can
reduce to $T\overline{T}$ or $T\overline{T}+\Lambda_{2}$ consistently. Using
the massive gravity method, we obtain the classically deformed action of a
sigma model of bosons and fermions interacting with an arbitrary potential,
extending previous results. As a consequence, a proposal regarding the
deformation of higher-derivative theories is made. Moreover, a standard
dimensional reduction procedure is presented, with the resulting operator
matching the form of prior findings under certain assumptions. In $d\geq2$, we
provide the exact structure of the quadratic terms in agreement with previous
approaches, as well as the structure of the linear and constant terms. All
higher order contributions are not easily evaluated, yet we derive the complete
answer for all cases up to seven dimensions. Under certain conditions, these
terms vanish, resulting in a quadratic operator. The trace-flow equation for
this family of deformations is also derived. Finally, we investigate the class
of root-$T\overline{T}$ operators in various dimensions within the scope of
this formalism.
