Supersymmetric gauge theories have been at the heart of research in theoretical physics for
the past ve decades. The study of these theories holds not only the promise for the ultimate
understanding of Nature, but also for discoveries of new mathematical constructions and
phenomena. The latter results from a highly geometrical nature of these theories, which is
prominently carried by the moduli space of vacua.
This thesis is dedicated to the study of moduli spaces of supersymmetric quiver gauge theories
with 8 supercharges. Typically, such moduli spaces consist of two branches, known as
the Coulomb branch and the Higgs branch. Following a review of the Higgs and the Coulomb
branch computational techniques in the rst part of this thesis, it is shown how the study
of Coulomb branches of 3d N = 4 minimally unbalanced theories is used for developing a
classification of singular hyperK ahler cones with a single Lie group isometry. As another application,
three-dimensional Coulomb branches are used to study Higgs branches of a stack of n
M5 branes on an A-type orbifold singularity. Analysis of such systems gives rise to a discrete
gauging phenomenon with importance for both physics and mathematics. From the physics perspective,
discrete gauging solves the problem of understanding the non-classical Higgs branch
phases of the corresponding 6d N = (1; 0) world-volume theory, even when coincident subsets
of M5 branes introduce tensionless BPS strings into the spectrum. From the mathematical
perspective, discrete gauging provides a new method for constructing non-Abelian orbifolds
with certain global symmetry. The thesis also includes an investigation of theories associated
with non-simply laced quivers. Remarkably, the formalized analysis of the so-called ungauging
schemes and the corresponding Coulomb branches reproduce orbifold relations amid closures
of nilpotent orbits of Lie algebras studied by Kostant and Brylinski. Finally, three-dimensional
Coulomb branches are employed to understand the Higgs mechanism in supersymmetric gauge
theories with 8 supercharges in 3; 4; 5, and 6 dimensions. It is illustrated that the physical
phenomenon of partial Higgsing is directly related to the mathematical structure of the moduli
space, and in particular, to the geometry of its singular points. The developed techniques provide
a new set of algorithmic methods for computing the geometrical structure of symplectic
singularities in terms of Hasse diagrams.