We initiate a systematic study of 2d(0,2) quiver gauge theories on the world-volume of D1-branes probing singular toric Calabi-Yau 4-folds. We present an algorithm for efficiently calculating the classical mesonic moduli spaces of these theories, which cor-respond to the probed geometries. We also introduce a systematic procedure for constructing the gauge theories for arbitrary toric singularities by means of partial resolution, which translates to higgsing in the field theory. Finally, we introduce Brane Brick Models, a novel class of brane configurations that consist of D4-branes suspended from an NS5-brane wrap-ping a holomorphic surface, tessellating a 3-torus. Brane Brick Models are the 2danaloguesof Brane Tilings and allow a direct connection between geometry and gauge theory.