  We review and extend the progress made over the past few years in
understanding the structure of toric quiver gauge theories; those which are
induced on the world-volume of a stack of D3-branes placed at the tip of a
toric Calabi-Yau cone, at an ``orbifold point'' in Kaehler moduli space. These
provide an infinite class of four-dimensional N=1 superconformal field theories
which may be studied in the context of the AdS/CFT correspondence. It is now
understood that these gauge theories are completely specified by certain
two-dimensional torus graphs, called brane tilings, and the combinatorics of
the dimer models on these graphs. In particular, knowledge of the dual
Sasaki-Einstein metric is not required to determine the gauge theory, only
topological and symplectic properties of the toric Calabi-Yau cone. By
analyzing the symmetries of the toric quiver theories we derive the dimer
models and use them to construct the moduli space of the theory both
classically and semiclassically. Using mirror symmetry the brane tilings are
shown to arise in string theory on the world-volumes of the fractional
D6-branes that are mirror to the stack of D3-branes at the tip of the cone.
