The Problem Of Gauge Theory

I sketch what it is supposed to mean to quantize gauge theory, and how this can be made more concrete in perturbation theory and also by starting with a finite-dimensional lattice approximation. Based on real experiments and computer simulations, quantum gauge theory in four dimensions is believed to have a mass gap. This is one of the most fundamental facts that makes the Universe the way it is. This article is the written form of a lecture presented at the conference Geometric Analysis: Past and Future (Harvard University, August 27-September 1, 2008), in honor of the 60th birthday of S.-T. Yau.