  We calculate the $\mathcal{S}$-multiplets for two-dimensional Euclidean
$\mathcal{N}=(0,2)$ and $\mathcal{N} = (2,2)$ superconformal field theories
under the $T\overline{T}$ deformation at leading order of perturbation theory
in the deformation coupling. Then, from these $\mathcal{N} = (0, 2)$ deformed
multiplets, we calculate two- and three-point correlators. We show the
$\mathcal{N} = (0,2)$ chiral ring's elements do not flow under the
$T\overline{T}$ deformation. Specializing to integrable supersymmetric seed
theories, such as $\mathcal{N} = (2,2)$ Landau-Ginzburg models, we use the
thermodynamic Bethe ansatz to study the S-matrices and ground state energies.
From both an S-matrix perspective and Melzer's folding prescription, we show
that the deformed ground state energy obeys the inviscid Burgers' equation.
Finally, we show that several indices independent of $D$-term perturbations
including the Witten index, Cecotti-Fendley-Intriligator-Vafa index and
elliptic genus do not flow under the $T\overline{T}$ deformation.
