  Geometric structures and dualities arise naturally in quantum field theories
and string theory. In fact, these tools become very useful when studying strong
coupling effects, where standard perturbative techniques can no longer be used.
In this thesis we look at several conformal field theories in various
dimensions. We first discuss the structure of the nilpotent networks stemming
from T-brane deformations in 4D $\mathcal{N}=1$ theories and then go to the
stringy origins of 6D superconformal field theories to realize deformations
associated with T-branes in terms of simple combinatorial data. We then analyze
non-perturbative generalizations of orientifold 3-planes (i.e. S-folds) in
order to produce different 4D $\mathcal{N}=2$ theories. Afterwards, we turn our
attention towards a few dualities found at strong coupling. For instance,
abelian T-duality is known to be a full duality in string theory between type
IIA and type IIB. Its nonabelian generalization, Poisson-Lie T-duality, has
only been conjectured to be so. We show that Poisson-Lie symmetric
$\sigma$-models are at least two-loop renormalizable and their
$\beta$-functions are invariant under Poisson-Lie T-duality. Finally, we review
recent progress leading to phenomenologically relevant dualities between
M-theory on local $G_2$ spaces and F-theory on locally elliptically fibered
Calabi-Yau fourfolds. In particular, we find that the 3D $\mathcal{N}=1$
effective field theory defined by M-theory on a local $Spin(7)$ space unifies
the Higgs bundle data associated with 4D $\mathcal{N}=1$ M-theory and F-theory
vacua. We finish with some comments on 3D interfaces at strong coupling.
