Over the last decade, we have witnessed remarkable progress in our understanding of Quantum Field Theories. New insights have emerged from a multitude of fronts, ranging from the Gauge/Gravity Correspondence to Integrability. In this seminar I will discuss Bipartite Field Theories (BFTs), a new class of QFTs embodying many of these new approaches. BFTs are 4d, N=1 quiver gauge theories with Lagrangians defined by bipartite graphs on Riemann surfaces. Remarkably, they underlie a wide spectrum of interesting physical systems, including: D-branes probing Calabi-Yau manifolds, their mirror configurations, integrable systems in (0+1) dimensions and scattering amplitudes in N=4 SYM. I will introduce new techniques for studying these gauge theories. I will explain how their dynamics is captured graphically and the interesting emergence of concepts such as Calabi-Yau manifolds, the Grassmannian and cluster algebras in the classification of IR fixed points. Finally, I will introduce a new framework for analyzing general systems of D3 and D7-branes over toric Calabi-Yau 3-folds. These ideas can be exploited for embedding BFTs in String Theory but have a much wider range of applicability.