  The holographic duality conjectures a relation between strongly coupled
quantum systems and quantum gravity in higher-dimensional spacetimes.
Gravitational theories in two and three dimensions are meaningful examples for
classical and quantum exploration due to their unique characteristics, notably
the absence of propagating bulk degrees of freedom and the presence of only
boundary degrees of freedom, distinguishing them from higher-dimensional
counterparts. These gravitational theories exhibit complex interactions when
the bulk spacetime has a finite size, regulated by Zamolodchikov's double-trace
irrelevant $T\overline{T}$ operator.
  This thesis aims to gain a holographic understanding of $\mathrm{AdS}_3$ and
JT gravity under the influence of the $T\overline{T}$ deformation. Under a
finite radial cutoff, these theories exhibit perturbative behavior that implies
the emergence of the Nambu-Goto action for the corresponding boundary graviton
action. We also conducted semi-classical calculations of observables related to
finite-cutoff gravity and its dual $T\overline{T}$-deformed CFT description,
including correlation functions involving stress tensors and gravitational
Wilson lines, along with an analysis of their supersymmetric extensions.
Additionally, we explored the implications of general stress tensor
deformations within field-theoretic and holographic settings.
  This thesis integrates previously adapted publications while also pioneering
new ground, notably exploring the definition of a quantum $T\overline{T}$
operator beyond two dimensions with $\frac{1}{N}$ corrections, investigating
quantum-corrected higher point correlators for a planar boundary, and offering
insights into a two-dimensional spherical boundary at a finite cutoff.
Furthermore, throughout the thesis, we show more details in calculations at
various points.
