  In this thesis we discuss how one can derive the quantum spectral curve for
the $\eta$-deformed AdS$_5 \times S^5$ superstring, an integrable deformation
of the AdS$_5 \times $S$^5$ superstring with quantum group symmetry. This model
can be viewed as a trigonometric version of the AdS$_5 \times $S$^5$
superstring, like the Heisenberg xxz spin chain is a trigonometric version of
the xxx spin chain. Our derivation starts from the ground-state thermodynamic
Bethe ansatz equations and discusses the construction of both the undeformed
and the $\eta$-deformed quantum spectral curve. We reformulate it first as an
analytic $Y$-system, and map this to an analytic $T$-system which upon suitable
gauge fixing leads to a $\mathbf{P}\mu$ system, the quantum spectral curve. We
then discuss constraints on the asymptotics of this system to single out
particular excited states. At the spectral level the $\eta$-deformed string and
its quantum spectral curve interpolate between the AdS$_5 \times $S$^5$
superstring and a superstring on "mirror" AdS$_5 \times $S$^5$, reflecting a
more general relationship between the spectral and thermodynamic data of the
$\eta$-deformed string. The thesis is set up such that it simultaneously
reviews the development of the undeformed AdS$_5 \times $S$^5$ quantum spectral
curve.
