  In this writing we shall address certain beautiful inter-relations between
the construction of 4-dimensional supersymmetric gauge theories and resolution
of algebraic singularities, from the perspective of String Theory. We review in
some detail the requisite background in both the mathematics, such as
orbifolds, symplectic quotients and quiver representations, as well as the
physics, such as gauged linear sigma models, geometrical engineering,
Hanany-Witten setups and D-brane probes.
  We investigate aspects of world-volume gauge dynamics using D-brane
resolutions of various Calabi-Yau singularities, notably Gorenstein quotients
and toric singularities. Attention will be paid to the general methodology of
constructing gauge theories for these singular backgrounds, with and without
the presence of the NS-NS B-field, as well as the T-duals to brane setups and
branes wrapping cycles in the mirror geometry. Applications of such diverse and
elegant mathematics as crepant resolution of algebraic singularities,
representation of finite groups and finite graphs, modular invariants of affine
Lie algebras, etc. will naturally arise. Various viewpoints and generalisations
of McKay's Correspondence will also be considered.
  The present work is a transcription of excerpts from the first three volumes
of the author's PhD thesis which was written under the direction of Prof. A.
Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of
MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac
vice; it is his sincerest wish that the ensuing pages might be of some small
use to the beginning student.
