We focus on supersymmetric gauge theories with eight superchrages in spacetime dimensions
d = 3, 4, 5, 6. These theories have very rich vacuum structures so our focus will be on their
moduli spaces of vacua. For d =, 4, 5, 6, we look at the Higgs branch moduli space. The
usual story is that the Higgs branch is a classical object that can be easily computed from
its Lagrangian. However, non-perturbative contributions can enhance the Higgs branch and a
classical description no longer works. In 6d N = (1, 0) and 5d N = (1, 0), these contributions
originate from tensionless BPS-strings and massless gauge instantons respectively as we tune
gauge coupling(s) to infinity. For 4d N = 2 theories, many gauge theories, and in particular
superconformal field theories (SCFTs), do not even have a Lagrangian description. We offer
a unifying solution to these problems in the form of magnetic quivers. These are 3d N = 4
gauge theories whose Coulomb branch is the same as the Higgs branch of the higher dimensional
theories. Using brane systems of D_d−D_(d+2)−NS5, with the possible inclusion of Od orientifold
planes, we show how the magnetic quivers of these theories can be extracted. Then, a) using
the monopole formula we study the moduli space as an algebraic variety by computing its
Hilbert series and b) using the new concept of Quiver subtraction we extract the phase diagram
(Hasse diagram) of these moduli spaces. Examples we explore include 5d SQCD theories at UV
fixed point, 4d rank one SCFTs, class S theories, S-fold theories etc. For the second outcome
of the thesis, we focus on new features of gauge theories with orthosymplectic gauge groups
such as discrete subgroups and non-simply laced edges, leading to a general classification of
such theories. For the final outcome, we study gauge theories with a mixture of unitary and
special unitary gauge groups which lead to a slew of new gauge theories related by 3d mirror
symmetry.