  Brane Tilings represent one of the largest classes of superconformal theories
with known gravity duals in 3+1 and also 2+1 dimensions. They provide a useful
link between a large class of quiver gauge theories and their moduli spaces,
which are the toric Calabi-Yau (CY) singularities.
  This thesis includes a discussion of an algorithm that can be used to
generate all brane tilings with any given number of superpotential terms. All
tilings with at most 8 superpotential terms have been generated using an
implementation of this method.
  Orbifolds are a subject of central importance in string theory. It is widely
known that there may be two or more orbifolds of a space by a finite group.
Abelian Calabi-Yau orbifolds of the form $\BC^3 / \Gamma$ can be counted
according to the size of the group $|\Gamma|$. Three methods of counting these
orbifolds will be given.
  A brane tiling together with a set of Chern Simons levels is sufficient to
define a quiver Chern-Simons theory which describes the worldvolume theory of
the M2-brane probe. A forward algorithm exists which allows us to easily
compute the toric data associated to the moduli space of the quiver
Chern-Simons theory from knowledge of the tiling and Chern-Simons levels. This
forward algorithm will be discussed and illustrated with a few examples. It is
possible that two different Chern-Simons theories have the same moduli-space.
This effect, sometimes known as `toric duality' will be described further. We
will explore how two Chern--Simons theories (corresponding to brane tilings)
can be related to each other by the Higgs mechanism and how brane tilings (with
CS levels) that correspond to 14 fano 3-folds have been constructed.
  The idea of `child' and `parent' brane tilings will be introduced and we will
discuss how it has been possible to count `children' using the symmetry of the
`parent' tiling.
