  We perform a detailed investigation of Bipartite Field Theories (BFTs), a
general class of 4d N=1 gauge theories which are defined by bipartite graphs.
This class of theories is considerably expanded by identifying a new way of
assigning gauge symmetries to graphs. A new procedure is introduced in order to
determine the toric Calabi-Yau moduli spaces of BFTs. For graphs on a disk, we
show that the matroid polytope for the corresponding cell in the Grassmannian
coincides with the toric diagram of the BFT moduli space. A systematic BFT
prescription for determining graph reductions is presented. We illustrate our
ideas in infinite classes of BFTs and introduce various operations for
generating new theories from existing ones. Particular emphasis is given to
theories associated to non-planar graphs.
