  In this thesis, we investigate properties of the one-parameter $\beta$- and
the three-parameter $\gamma_i$-deformed descendents of $\mathcal{N}=4$ SYM
theory. We find additional multi-trace interactions that must be included in
the deformed theories to prevent persistent divergences in observables. For the
$\gamma_i$-deformation, we show in an explicit Feynman-diagrammatic calculation
that these interactions have running coupling constants which spoil the
conformal invariance of the quantised theory, even in the 't Hooft limit. We
show that the running coupling constant in principle also affect the anomalous
dimensions of composite operators by calculating the $K=L$ loop leading order
wrapping corrections to the operators tr$\bigl(\phi_i^L)$. For the
$\beta$-deformation, we analyse the impact of multi-trace couplings on the
one-loop spectrum and determine the complete one-loop dilatation operator of
the conformal $\beta$-deformation in the 't Hooft limit. Based on the
field-theoretic data from the $\beta$- and $\gamma_i$-deformation, we propose a
test to determine whether supersymmetry and/or exact conformal invariance are
necessary prerequisites of the quantum integrability found for $\mathcal{N}=4$
SYM theory. We also generalise the P\'{o}lya-theoretic approach to the thermal
one-loop partition function to be also applicable in the setting of the
$\beta$- and $\gamma_i$-deformation. We find that the deconfinement
phase-transition temperature in the deformed theories is the same as in the
undeformed theory at one-loop level and we conjecture that it remains the same
even non-perturbatively. Finally, we also provide many technical details
concerning Feynman-diagram techniques and present the tool FokkenFeynPackage
which implements these rules into Mathematica.
