Quantum Yang-Mills theory describes the interactions of the quarks and gluons that are the microscopic constituents of matter. In perturbation theory, quantum Yang-Mills theory is mathematically well defined. It is, however, very difficult to compute with, and many basic properties remain conjectural.
I will first review some of the mathematics and physics of quantum Yang-Mills theory, and if quantum field theory more generally. Then I will explain how there is a hidden integrability in a self-dual subsector of certain Yang-Mills theories. In this subsector, there is a large quantum group symmetry algebra which allows for efficient computation of physically relevant quantities.