In recent years, a complete reformulation of perturbative quantum field theory has been found for certain theories. Although a similar reformulation is likely to exist for all theories, it is exceptionally simple in the case of planar, maximally supersymmetric Yang-Mills. I will briefly review this new formulation, which recasts the S-matrix in terms of a new class of field-theoretic objects called `on-shell diagrams.' I will describe how on-shell diagrams are connected with what is known as the `positroid' stratification of the Grassmannian, and will illustrate how they can be characterized and computed combinatorially. I will illustrate the power of these new tools, and describe some of the recent surprises they have uncovered in the structure of multi-loop amplitudes.