In the past ten years, the study of scattering amplitudes in quantum field theory has led to a revolutionary reformulation of the subject. This revolution began with the discovery, in 2005, of a recursive expansion for scattering amplitudes (to leading order) in terms of planar, trivalent, two-colored graphs---called "on-shell diagrams." Around the same time that these diagrams were first drawn by physicists, they also started to appear in the mathematical literature (for entirely independent reasons) in the context of what is known as the "positroid stratification" of Grassmannian manifolds. Recently, these two independent lines of research came together, leading to many valuable insights on both sides. In my talk, I will outline the physical ideas behind these developments, and explain the many deep connections which have been found between scattering amplitudes and the geometry and combinatorics of the positroid stratification of the Grassmannian.