Many of the most profound insights into the simplicity, structure, and mathematical form taken by observables in perturbative quantum field theory were discovered after heroic feats of computation. These discoveries have, in turn, led to dramatic advances in our ability to use quantum field theory to connect theory to experiment; and this has in turn led to even further discoveries. This virtuous cycle continues unabated, leading to a profoundly more powerful description of the basic, mathematical machinery involved in quantum field theory. I review this progress, highlight the challenges confronting us today, and outline a number of recent insights that promise to dramatically improve the way we make computations going forward.